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I have little to no substantial experiences with sums at infinity beyond what the notation conveys.

My question is how does one calculate what: $$\sum_{n=1}^{\infty} a(n) $$ converges to, supposing the sum to be convergent. To use a classic example, what general methods could be used to calculate what $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ converges to. I am aware that it converges to $\frac{\pi^2}{6}$. I am aware that this is proven via many methods, but using general introductory calculus methods is it possible? If not why, etc.

My apologies if this is vague. What methods should I look up/research, are there any general forms which are tedious but work most of the time, etc.

Something I have heard in passing is the integral representation of a sum, what is this, how is it derived, and can\is it generalized or question specific?

I don't need these all answered at once, if you know one of the answers but not all that is just fine! :)

halrankard
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Euler Wannabe
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    You need different methods for different sums, and, most convergent sums won't actually have a closed form. That particular sum is the Basel problem https://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-limits-k-1-infty-frac1k2-basel-pro –  Aug 20 '20 at 03:46
  • What do you mean by closed form? – Euler Wannabe Aug 20 '20 at 03:48
  • Essentially that you cannot find the solution exactly in terms of 'classical' functions. For example, what does $\sum_{n=1}^\infty \frac{\sin n}{n^2}$ converge to? If you find an exact answer to that question you should publish a paper. –  Aug 20 '20 at 03:51
  • Its 42 obviously :), in seriousness, what is an example of a classical function in this context, I have had not much exposure to this. – Euler Wannabe Aug 20 '20 at 03:55
  • *and how do you calculate a classical functions \sum convergence – Euler Wannabe Aug 20 '20 at 03:56
  • Well, I am not entirely sure what I mean. Let us take that sum for example. We could easily say it converges to $E_w$ (for Euler Wannabe's number) and is defined as $E_w:=\sum_n\frac{\sin n}{n^2}$. That number exists and can be calculated to equal accuracy as $\pi$. But it doesn't have any meaning. If you were able to express $E_w$ in terms of other numbers (e.g. rational numbers, $\pi$, $e$, etc) or show it to have some fundamental geometric meaning than that is a very interesting result. I guess my point is that $E_w$ does not necessarily have any relation to numbers we may call 'classical'. –  Aug 20 '20 at 04:03
  • So my question is what methods could we use to calculate it exactly? If I am understanding you right you are saying that sin(n)/n^2 converges to E_w which is a defined (irrational I think) number which we can calculate, so what methods could we use to calculate lets say 5 digits of E_w, or am I misunderstanding. – Euler Wannabe Aug 20 '20 at 04:07
  • *one of my questions :) – Euler Wannabe Aug 20 '20 at 04:09
  • You can calculate it approximately, just take as many terms from the sum as you like. I would bet my house that it is irrational and transcendental, but I wouldn't have a clue how to prove it. My point is that there does not necessarily exist a method to calculate it exactly (whatever that means), and if there does it will be particular to that sum and not necessarily applicable to any other sum. –  Aug 20 '20 at 04:11
  • Hmmm, when it comes to proving a sum convergence to a transcendental number what methods in the past have been used, even if they are case specific, also on the subject what are the various methods that are used when it converges to a non-transcendental number? I am thinking approximately Calculus II level methods if I am not mistaken. Finally how does one prove convergence in the latter case. From what I have gathered this is a pretty wide topic so I wanted to try and understand the basics first. – Euler Wannabe Aug 20 '20 at 04:19

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