## Some random thoughts on elementary calculus

Here is a quick and very simple observation about elementary sums and integrals, summation by parts and Fubini.

Let us compare

$\sum_{n=1}^{\infty} \frac{n}{2^n}$ and

$\int_1^{\infty} \frac{x}{2^k} dx$.

There are many ways to handle the sum above, but perhaps the most straightforward is the following. Rewrite the sum in the form

$\sum_{n=1}^{\infty} 2^{-n} \left\{ \sum_{k=1}^n 1 \right\}$.

Changing the order of summation we obtain

$\sum_{k=1}^{\infty} \sum_{n=k}^{\infty} 2^{-n}$ and the result is easily obtained by summing up the geometric series twice.

We just applied the summation by parts principle. Nevertheless, we get an interesting perspective on things when we apply the same idea to the integral. To make things a bit clear, let’s replace $2^{-x}$ by $e^{-x}$. We have

$\int_1^{\infty} \frac{x}{e^x} dx$

$\int_1^{\infty} e^{-x} \int_0^x dy dx$.

Reversing the order of integration we obtain

$\int_1^{\infty} \left\{ \int_y^{\infty} e^{-x} dx \right\} dy$

$=\int_1^{\infty} e^{-y} dy=e^{-1}$.

This process is, of course, equivalent to the integration by parts, but it is amusing nevertheless. In general, consider

$\int_a^b f'(x)g(x)dx$

$\int_a^b f'(x) \left\{ \int_a^x g'(y)dy+g(a) \right\} dx$

$=(f(b)-f(a))g(a)$

$+\int_a^b \int_a^x f'(x)g'(y) dydx$

$=(f(b)-f(a))g(a)+\int_a^b \left\{ \int_y^b f'(x) dx \right\} g'(y)dy$

$(f(b)-f(a))g(a)+\int_a^b (f(b)-f(y))g'(y)dy$

$(f(b)-f(a))g(a)+(g(b)-g(a))f(b)-\int_a^b f(y)g'(y)dy$

$=g(b)f(b)-g(a)f(a)-\int_a^b f(y)g'(y)dy$, which is just the classical integration by parts formula derived using the fundamental theorem of calculus and Fubini.

## Princeton and President Wilson

Reuters is reporting that Princeton may scrub U.S. President Wilson’s name for racist ties. There is absolutely no doubt that President Wilson was a racist. He reintroduced segregation for Federal workers. He also told a group of visiting African American leaders that segregation is a good thing and then kicked them out. There is a long list of other actions of this type, but they only serve to reinforce and already obvious point. And yet, several interesting questions immediately arise. How far are we prepared to go to condemn offensive views of our ancestors? Just as importantly, are we truly prepared to be consistent in this process?

Let us begin by going further back in time to 1958, when Abraham Lincoln addressed a crowd in Springfield, Illinois, in the last speech of the 1858 Senate campaign. Here is the part of speech I find most interesting: ” … Through all, I have neither assailed , nor wrestled with any part of the Constitution. The legal right of the Southern people to reclaim their fugitives I have constantly admitted. The legal right of Congress to interfere with their institution in the states, I have constantly denied. In resisting the spread of slavery to new teritory, and with that, what appears to me to be a tendency to subvert the first principle of free government itself my whole effort has consisted. To the best of my judgment I have labored for, and not against the Union. As I have not felt, so I have not expressed any harsh sentiment towards our Southern bretheren. I have constantly declared, as I really believed, the only difference between them and us, is the difference of circumstances.”

There is no real comparison between Lincoln’s position and that of Wilson, and yet it is difficult to stomach the words “The legal right of the Southern people to reclaim their fugitives I have constantly admitted”, or “…the only difference between them and us, is the difference of circumstances.” Would Lincoln really phrase things in this way if the South had enslaved white Methodists? And there are many other quotes by Lincoln that raise serious questions as well. He once said, ““My paramount object in this struggle is to save the Union, and is not either to save or destroy slavery.”

He also said, ““…I will say in addition to this that there is a physical difference between the white and black races which I believe will forever forbid the two races living together on terms of social and political equality.” There is more, but the point is clear enough. Are we prepared to take down the Lincoln memorial in view of these horrific and morally repugnant statements?

The question of consistency brings us back to the modern times. President Obama, widely acknowledged as an enlightened and progressive President, sat through numerous anti-Semitic sermons by Reverend Jeremiah Wright and, to the best of my knowledge, never complained. Obama has frequently said that Reverend Wright is one of the major influences in his life. Reverend Wright continued making blatantly anti-Semitic statements throughout Obama’s presidency and Obama did not choose to comment on any of them. Can future generations, in good conscience, put Obama’s names on college campus buildings?

It is also important to address the issue of hypocrisy from a slightly different point of view. I do not have specific data on the folks who occupied the office of the Princeton President in order to address the issue of President Wilson, but I wonder how they feel about the well-documented rise in anti-Semitism the Harvard campus. The problem was identified back in 2002 by Larry Somners (http://www.nytimes.com/2002/09/21/us/harvard-president-sees-rise-in-anti-semitism-on-campus.html) and things have only gotten worse since then. What do these protesters think about schools like Columbia providing a forum for President Ahmadinejad, a virulent anti-Semite, and yet violently protesting speeches by Alan Dershowitz, a life-long supporter of human rights?

In short, I am of two minds on the topic of removing President Wilson’s name from Princeton’s buildings. On one hand, I am happy to see a serious examination of racism, misogyny and homophobia in our nation’s past and present. On the other hand, this process can quickly degenerate into an indiscriminate witch hunt where a balanced and critical understanding of the past will not prevail. Moreover, too often this mission is led by hypocritical activists who are trying to establish the hegemony of a rather narrow spectrum of views under the cover of progressive and inclusive ideology.

## The Tsarnaev verdict: bloodlust and excuses

The federal jury decided yesterday that the proper way to mete out justice in the Dzhokhar Tsarnaev case is to execute the 21 year old who, along with his brother, Tamerlan, was the junior member of the duo that set off a bomb that killed three people and injured many others during the Boston Marathon on April 15, 2013. Dzhokhar tried to kill himself during the stand-off with the police by shooting himself threw the jaw. After a lengthy hospital stay, he was put on trial and easily convicted of multiple homicides among other charges. A life sentence without a possibility of parole could have followed and Dzhokhar Tsarnaev would have joined the large and rather stinky proverbial trash bin of history. But the jurors in perhaps the most progressive State of the Union, Massachusetts, decided to impersonate a lynch mob from a 19th century Old West town of your choosing and convicted Dzhokhar to death, citing his lack of remorse and a heinous nature of his crimes.

Reasonable people are going to tell me that calling the federal jury a lynch mob is a hyperbole designed to stir up emotions and fears. I disagree. The feature of the lynch mob that people typically focus on is its extra-judicial nature. But this does not go nearly far enough, especially in the day and age when even the most oppressive and dictatorial regimes in the world figured out how to use officially setup courts as fig leaves for their activities. There is no evidence that the Tsarnaev jury was manipulated by the government, but it was certainly influenced by the public opinion. It turns out that liberalism in Massachusetts quickly reaches its limitations when the inhabitants of Boston are threatened on their own streets. What is far more disturbing is that the blood lust did not stop long after it became clear that the Boston Marathon bombing was an isolated event and not a part of a systematic campaign to terrorize our nation. What makes the Tsarnaev jury a lynch mob is that it made the decision to execute Dzhokhar based on fear and mass hysteria, not on reason and fundamental interests of our society.

Let us take a few steps back in time and think back to World Trade Center attack on September 9, 2001. I remember very clearly discussing the events with friends and family and noting that the degree of success of this outrageous crime will be determined by our reaction. As terrible as the attack was, the overall cost, both interns of human casualties and infrastructure damage was relatively small compared to what happened after. I do not believe for a moment that Bin Laden and his goons would have been satisfied if the story of 9/11 ended on that day. What made it the most successful terrorist attack in history is the subsequent invasion of Iraq and Afghanistan which undermined American standing in the world, damaged our financial infrastructure and endangered our national security for generations to come.

This essay is not written by a pacifist. I am all for the use of force when terrorists pose a clear and present danger. But in the case of Tsarnaev, execution does nothing more than fulfill our carnal need for vengeance. It will do nothing to deter other terrorists. Does anybody really believe that life imprisonment is preferable to a typical jihadist who is seeking glory in afterlife? Putting this guy away for life quietly and without fanfare would have been much more effective.

As the impact of the Tsarnaev death verdict is slowly sinking in, excuses abound. Every possible explanation for the verdict has been tried. Some have argued that death penalty is a statement that our society does not tolerate terrorism. Others have pointed out that dead Tsarnaev cannot be traded for kidnapped hostages. Historical precedents where massive use of the death penalty allegedly deterred terror have been brought up as well. I do not necessarily doubt the motivation of the authors of these arguments. But the fundamental reason we are going to kill Tsarnaev is that he injured us and we want revenge. The rest is simply window dressing.

studentviews

## Math171: Some calculations involving elementary limits

I am teaching honors calculus again in the Fall and I am very happy about that. Even though the first class is many months away, I am already thinking about how I am going to present various aspects of the material. In my experience, no matter how clearly one thinks he is presenting $\delta-\epsilon$ proofs, it takes a while for students to absorb the techniques and use them in a proper context. Here is another attempt to explain the basics of this subject matter clearly.

We say that $\lim_{x \to a} f(x)=L$ if given $\epsilon>0$ there exists $\delta>0$ such that $|f(x)-L|<\epsilon$ whenever $|x-a|<\delta$. But what does this actually mean? When we say that $\lim_{x \to a} f(x)=L$, we are using mathematical notation to express the idea that $f(x)$ gets closer and closer to $L$ as $x$ gets closer and closer to $a$.

How do we make this intuitive idea mathematically precise? Being able to get arbitrarily close to $L$ means that the difference between $f(x)$ and $L$ can be made as small as wish. Let us give the quantity “as small as we wish” a name and call it $\epsilon$. This gets us started and tells that we wish for $f(x)$ to be within $\epsilon$ of $L$. But this is presumably going on while $x$ is getting closer and closer to $a$. What does that mean?

Before going further, let us consider an example. Let $f(x)=5x$ and $a=0$. Then the statement $\lim_{x \to a} f(x)=L$ takes the form $\lim_{x \to 0} 5x=0$. I am sure that we all believe that indeed $5x$ is getting closer and closer to $0$ as $x$ gets closer and closer to $0$.

The first thing we do is decide how close we want our function $f(x)$ to be to limit $L$. This “how close” is typically measured using the quantity called $\epsilon$. So we want to be within $\epsilon$ of $L$, which in our case equals to $0$. For $5x$ to be within $\epsilon$ of $0$, we must have

$|5x|<\epsilon$, which is equivalent to the statement

$|x|<\frac{\epsilon}{5}$.

We can put everything together in the following way. Let $\epsilon>0$ be given. Let $\delta=\frac{\epsilon}{5}$. Then $|5x|<\epsilon$ whenever $|x|<\delta$.

Let’s try a more elaborate example. Let’s compute

$\lim_{x \to 1} x^2+x-1$.

A quick glance tells us that the answer should be $1$ because $1^2+1-1=1$ and there are no apparent pitfalls. We need to make this observation formal, so let’s go for it.

Here $f(x)=x^2+x-1$, $a=1$ and $L=1$. We want to be within a given threshold $\epsilon$ of $1$. So we want

$|x^2+x-1-1|<\epsilon$.

Observe that

$x^2+x-2=(x-1)(x+2)$.

It bears repeating, with this observation in mind, that we want

$|(x+2)(x-1)|<\epsilon$ whenever

$|x-1|<\delta$, where $\delta$ is chosen depending on $\epsilon$.

If in place of $(x+2)(x-1)$ we had something like $5(x-1)$, we would know exactly what to do and set $\delta=\frac{\epsilon}{5}$ like before. But what do we do with the factor of $x+2$?

Observe that

$|x+2|=|x-1+3| \leq |x-1|+3.$

This means that if $|x-1|<\delta$, then

$|x+2|<\delta+3$.

If we insist that, among other constraints, $\delta \leq 1$, then

$|x+2|<4$.

Let us now connect the dots. Let $\epsilon>0$ be given. Let

$\delta=\min \{1, \frac{\epsilon}{4} \}$.

Since $\delta \leq 1$, $|x+2| \leq 4$, as we so above. Therefore

$|f(x)-L|=|(x-1)(x+2)|<4 \delta \leq \epsilon$ by definition of $\delta$. This completes the proof.

How does one make sure that what is written above makes sense? Just work a couple of similar examples. For instance, prove that

$\lim_{x \to 2} x^2+x+3=9$.

Let us now consider a slightly more complicated example. We shall prove that

$\lim_{x \to 1} x^3=1$.

This is the same as proving that

$\lim_{x \to 0} x^3-1=0$ and using the experience above as a guide we factor

$x^3-1=(x-1)(x^2+x+1)$.

This is done by computing the quotient $\frac{x^3-1}{x-1}$. If you do not remember how to divide polynomials, please take a look at http://www.purplemath.com/modules/polydiv2.htm for a very nice and straightforward review of this important procedure.

Going back to our limit problem, given $\epsilon>0$ we must find $\delta>0$ such that $|x^3-1|<\epsilon$ whenever $|x-1|<\delta$. Equivalently, we need

$|x-1||x^2+x+1|<\epsilon$ whenever

$|x-1|<\delta$.

With the previous example as our guide, we look for an appropriate upper bound for $|x^2+x-1|$ keeping in mind that $|x-1|<\delta$.

To do this, we write $x^2+x+1$ as a polynomial in $x-1$. More precisely, we notice that

$x^2+x+1={(x-1)}^2+3(x-1)+3$.

It follows that

$|x^2+x+1|<\delta^2+3 \delta+3$.

If we insist, as before, that $\delta \leq 1$, then

$|x^2+x+1|<7$ since $\delta^2 \leq \delta$ if $\delta \leq 1$.

We are now ready to put everything together. Let $\epsilon>0$ be given. Choose $\delta=\min \{1, \frac{\epsilon}{7} \}$. Then

$|x^3-1|=|x-1||x^2+x+1|<7 \delta \leq \epsilon$. This completes the proof.

Once again it is important to cement this idea and this can be done, for example, by proving that

$\lim_{x \to 2} x^3+x+4=14$.

A simple remark is in order here. Regardless of what polynomial we have on the left hand side, we know that if the limit is what we claim it is, then

$\lim_{x \to 2} x^3+x+4-14=0$ and this means that we must be able to divide $x^3+x-10$ by $x-2$. This is a consequence of a fact, which we are going to review in class, that if $f(x)$ is a polynomial and $f(a)=0$, then

$f(x)=(x-a)g(x)$, where $g(x)$ is a polynomial of one degree less.

Let us consider one more example where the main complication is to avoid accidentally dividing by $0$. Consider

$\lim_{x \to 1} \frac{1}{x}$.

We all believe that the limit is equality $1$, I am sure, and the trick is to carry out the $\delta-\epsilon$ argument in such a way that the fact that $\frac{1}{x}$ gets tricky as $x$ gets close to $0$ does not get in the way.

We want to make $|\frac{1}{x}-1|$ smaller than $\epsilon$ if $|x-1|<\delta$. This is the same as showing that

$\frac{|x-1|}{|x|}<\epsilon$ whenever

$|x-1|<\delta$.

With the previous examples as our guides, we know that the game is to argue that we may reduce to the case where we get to bound $\frac{1}{|x|}$ by some constant $C>0$. We will then be able to set

$\delta=\frac{\epsilon}{C}$. But what is $C$ in  this case and how do we go about running the procedure? The enemy is the situation when $|x|$ is close to $0$. The point is that if $|x-1|$ is small enough, then $|x|$ cannot possibly be very close to $0$. For example, we may insist that $|x-1|<\frac{1}{2}$. This implies that $|x|>\frac{1}{2}$ and we should be fine since $\frac{1}{|x|}$ is now guaranteed to be $<2$ and we can take $\delta=\frac{\epsilon}{2}$.

Let us now put everything together. Let $\epsilon>0$ be given. Set

$\delta=\min \{\frac{1}{2}, \frac{\epsilon}{2} \}$. It follows that

$|\frac{1}{x}-1|=\frac{|x-1|}{|x|}<2|x-1|<2 \delta \leq \epsilon$ and the proof is complete.

In order to make sure that this calculation is clear, prove that

$\lim_{x \to 3} \frac{1}{x-\frac{1}{5}}=\frac{4}{15}$.

## President Roosevelt and the Holocaust: New Evidence

By Dr. Rafael Medoff

Fearful of Jewish ‘domination’ and ‘overrepresentation,’ his vision of America did not allow for too many Jews.
“During my research I found numerous examples of behind-the-scenes remarks in which U.S. president Franklin Delano Roosevelt spoke about the danger of allowing Jews in large numbers to live in one specific place, or to become too prominent in various professions.

“He also promoted imposing a quota on the admission of Jewish students to Harvard in the 1920’s.

“In 1943, Roosevelt urged local leaders in Allied-liberated North Africa to limit the entry of Jews into many professions. He claimed that ‘the complaints which the Germans bore towards the Jews’ were ‘understandable’ because there were many Jews in law, medicine, and other fields in Germany.”

Dr. Rafael Medoff is founding director of The David S. Wyman Institute for Holocaust Studies, a research and public education institute based in Washington, D.C. In 2013 he authored his fourteenth book – “FDR and the Holocaust: A Breach of Faith”.

He remarks: “In 1943, Roosevelt endorsed a plan by one of his senior advisers to ‘spread the Jews thin all over the world’ so they would quickly assimilate. He also claimed, in 1938, that the Jews were too prominent in Poland’s economy, suggesting that this was the cause of anti-Semitism there. This helps explain why Roosevelt refused to allow Jewish refugees to enter the U.S. up to the limit of the existing laws. Fearful of Jewish ‘domination’ and ‘overrepresentation,’ his vision of America did not allow for too many Jews.

“Roosevelt was well known for following public opinion rather than leading it. He knew that it would have been unpopular to propose liberalizing America’s immigration quotas. Yet changing the quotas was not necessary to save Jewish refugees. All Roosevelt had to do was quietly tell the State Department – which was in charge of immigration – to allow the existing quotas to be filled. This would have been in accordance with the law, so opponents would have been hard-pressed to muster a serious argument against it.

“There were 190,000 unused quota places from Germany and Axis-occupied countries during the Hitler years. The annual quota from Germany – about 26,000 until 1938, 28,000 thereafter – was filled during only one of Roosevelt’s twelve years in office. Most other years, it was less than one-quarter filled.

“Roosevelt refused to support the Wagner-Rogers bill of 1939, which would have permitted the entry of 20,000 German Jewish children outside the quota system. Those children would not have taken away any jobs, an argument often heard regarding allowing more immigrants in. Yet only one year later, Roosevelt personally intervened to enable thousands of British children to come to America to escape the German blitz of London.

“Roosevelt could have done other things which would have saved victims of the Holocaust.

-“He could have pressured the British to open Palestine’s doors to Jewish refugees.

-“He could have authorized the use of empty troop supply ships to bring refugees to the U.S. temporarily, until the end of the war.

-“Roosevelt could have permitted refugees to stay as tourists in a U.S. territory such as the Virgin Islands, until it was safe for them to return to Europe.

-“He could have also authorized the bombing of Auschwitz or the railway lines leading to it, which would have interrupted the mass-murder process.

“The issue of the failure to bomb Auschwitz never seems to go away, because in many ways it sums up America’s refusal to make even a minimal effort to interrupt the mass-murder process. U.S. planes were flying just a few miles from the gas chambers, hitting German oil factories. Yet the administration never ordered them to drop a few bombs on the death camp.

“Previous research indicated that these requests were rejected by the Assistant Secretary of War and lower level officials. My book shows, for the first time, that those requests in fact reached all the way to Roosevelt’s most important cabinet members, Secretary of State Cordell Hull and Secretary of War Henry Stimson. But they did not act.

“This issue is relevant today. Every American president faces the question of whether to use U.S. military force on behalf of humanitarian objectives in other parts of the world. President Clinton eventually intervened in Bosnia and says he regrets not having intervened in Rwanda. Neither President Bush nor President Obama intervened in the Darfur genocide. Obama did act in Libya, but has been much more hesitant regarding Syria’s slaughter of its citizens and Iran’s genocidal threats against Israel.

“The reasons for the failure to bomb Auschwitz, and the ways in which the Roosevelt administration misled the groups requesting such bombing, offer many important lessons for dealing with today’s global problems.”

Medoff concludes: “Roosevelt deserves credit for lifting America out of the Great Depression and his leadership in World War II, but he was not the humanitarian and champion of ‘the forgotten man’ that he claimed to be – at least not when it came to the Jews.”

## Math171: A remark on mathematical induction

Let us consider some examples of some simple statements that we can prove directly or by using mathematical induction. Let us begin with the assertion

$1+2+3+\dots+n=\frac{n(n+1)}{2}$.

Let us first give a direct proof. Take $n^2$ dots arranged in an $n$ by $n$ grid. Mark the dots on the diagonal running from the bottom left corner to the upper right corner with a red marker. Mark the dots above the diagonal with a blue marker and those below by a green marker.

We are now ready to go to work. The total number of dots in our grid is $n^2$. The number of red dots is exactly $n$. The number of blue dots is exactly the same as the number of green dots.

Now observe that the number of blue dots (or green dots) plus the number of red dots is equal to

$1+2+3+\dots+n$.

This is because the number of blue dots in the uppermost row is $n$. The number of blue dots one row below is $n-1$, and so on. It follows that

$n^2=RED+BLUE+GREEN=RED+2 \cdot BLUE=n+2 \cdot BLUE$

We conclude that

$BLUE=\frac{n^2-n}{2}$ and hence

$1+2+\dots+n=BLUE+RED=\frac{n^2-n}{2}+n=\frac{n^2+n}{2}=\frac{n(n+1)}{2}$.

This completes the direct proof and we now turn our attention to the proof by induction. First of all, the statement is true when $n=1$ since

$1=\frac{1(1+1)}{2}$.

Assume the statement holds for $n-1$. This means that we assume that

$1+2+\dots+n-1=\frac{(n-1)n}{2}.$

Add $n$ to both sides of the equation. We get

$1+2+\dots+n=\frac{(n-1)n}{2}+n=\frac{n^2-n+2n}{2}=\frac{n^2+n}{2}=\frac{n(n+1)}{2}$, as desired and this completes the proof.

An important point is that while the proof by induction is simpler, it requires us to be able to guess in advance that $1+2+\dots+n=\frac{n(n+1)}{2}$. The direct proof, while more complicated, actually derives the expression for $1+2+\dots+n$ from scratch.

Let us a dig a bit deeper. Suppose that RED-BLUE-GREEN argument did not occur to us. What else might we have thought of? Let us write

$1+2+\dots+n=\sum_{j=1}^n j$.

If you are fuzzy on the summation notation, here is the basic idea. The expression $\sum_{j=1}^n a_j$ equals

$a_1+a_2+a_3+\dots+a_n,$ where $a_j$s are real numbers. For example, if $a_j=j^2$,

$\sum_{j=1}^n a_j=\sum_{j=1}^n j^2=1^2+2^2+\dots+n^2$.

Note that $j$ is what is often called a “dummy variable” because putting a different letter in does not change the meaning of the sum. For example,

$\sum_{m=1}^n m^2=1^2+2^2+\dots+n^2$, just as above.

Just to make sure we understand how everything works, let’s change the bounds of summation. We have

$\sum_{j=5}^N j^2=5^2+6^2+\dots+N^2$, and, in a more concrete situation

$\sum_{j=3}^5 j^2=3^2+4^2+5^2=50$.

Going back to the problem at hand, we are considering

$1+2+\dots+n=\sum_{j=1}^n j$.

Even if we cannot guess the answer right away, we may be able to guess a “form” of the solution. For example, let us take a step back and look at an even simpler situation. Consider

$1+1+\dots+1$, $n$ times. The answer is, of course $n$. We can represent this sum as

$\sum_{j=1}^n 1.$

Recall that a polynomial of degree $k$ in the variable $x$ is the expression of the form

$b_0+b_1x+b_2x^2+\dots+b_kx^k$.

Observe that $1$ is a polynomial of degree $0$ in the variable $n$, and $n$ is a polynomial of degree $1$ in the variable $n$, so when we say that

$\sum_{j=1}^n 1=n$, we are saying, roughly, that summing a polynomial of degree $0$ yields a polynomial of degree $1$. Mathematics is all about guessing and conjecturing so we now let our imagination take over and see what happens. If summing a polynomial of degree $0$ yields a polynomial of degree $1$, is it such a huge leap to guess that summing a polynomial of degree $1$ should yield a polynomial of degree $2$? Let’s try it!

We have $\sum_{j=1}^n j$ and since we are summing a polynomial of degree $1$, we are going to guess that we are going to end up with a polynomial of degree $2$. Every polynomial of degree $2$ in the variable $n$ is of the form $an^2+bn+c$, where $a,b,c$ are real numbers. So our guess is that

$\sum_{j=1}^n j=an^2+bn+c.$

Note that we can start these sums at $0$ instead of $1$ and the value remains unchanged. This shows that summing up to $0$ yields $0$ as the answer, so $c=0$, so we are down to examining $an^2+bn$. If $n=1$,

$1=a \cdot 1^2+b \cdot 1$, which tells us that $a+b=1$.

If $n=2$, we have

$1+2=a \cdot 2^2+b \cdot 2$, so

$3=4a+2b$.

This leads us to a system of equations

$a+b=1; \ 4a+2b=3.$

Solving this system yields $a=b=\frac{1}{2}$, which puts our guess in the form

$\sum_{j=1}^n j=\frac{n^2+n}{2}=\frac{n(n+1)}{2}$.

Note that what we just wrote is not a proof, but it was a way of obtaining a PLAUSIBLE expression for $\sum_{j=1}^n j$. Once we have such a plausible expression, we can try to prove it by induction. In this case, the induction will work since we have already seen above that our “guess” is actually correct.

An interested reader should carry out this idea of “guessing” and then verifying by induction in the case of the sum

$\sum_{j=1}^n j^2=1^2+2^2+\dots+n^2$.

If the line of reasoning above generalizes, one should guess that this expression is a polynomial of degree $3$ in $n$. In other words,

$\sum_{j=1}^n j^2=an^3+bn^2+cn+d$.

Once again, it is not difficult to argue that $d=0$. Please go ahead and figure out what $a,b,c$ must be and verify your answer using mathematical induction. Let me know what happens!