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In the 10 tests of convergence/divergent (that I know), them being,

by defn, integral test, div test, comparison test, limit comparison test, gs test, alternating series test, p-series test, root test, ratio test

the only ones that can tell me what the series converges to is by defn and the gs test.

The other ones can tell me if the series converges or diverges.

But I've always wondered, why do we care whether it converges, if we cannot exactly figure out what it converges to? Why is it helpful to know only the behaviour and not the actual value?

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    $\begingroup$ Of course we care about both, but in practice it's generally a lot easier to decide convergence than to actually compute the sum. A good convergence result often comes with information about the speed of convergence...so that we can use it to compute the series numerically, which is often the best we can manage. $\endgroup$ – lulu Apr 20 '17 at 16:58
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    $\begingroup$ There are ways (in specific cases) to prove that if a sequence converges, then it must converge to $L$. For example, some recurrences, if they converge, converge to a fixed point. Also, for many of the tests, you can get estimates on the remaining error. $\endgroup$ – Michael Burr Apr 20 '17 at 16:59
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    $\begingroup$ You can numerically compute the sum and see it goes to a certain approximate value, but what if it diverges slowly? The most dangerous and famous example is the harmonic series, which diverges but really slowly. So, knowing if it converges in the first place is essential. $\endgroup$ – AspiringMathematician Apr 20 '17 at 17:10
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    $\begingroup$ Well, a professor of mine once said "Before you go digging around in the mud, you want to know that there's a potato in there". In context it depends on what you want to do. If the goal is not to actually solve an equation but just to understand a behavior--- say, you want to know if a function is bounded or not. If it is bounded/unbounded you can say such and such will happen. We just want to know the behavior, we don't care what the bound itself is nescessarily. Except when we do. $\endgroup$ – fleablood Apr 20 '17 at 17:15
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Here are some ideas:

  • Tests like the integral test actually include bounds on the error. For example, if the integral test applies to $\sum a_n$, then you could compute $\sum_{n=1}^Na_n$ and use the integral test on the remainder to bound the remainder.

  • The alternating series test also comes with a bound on the error, so you know how close your partial sum is to the right answer (if you were working on a computer, for example).

  • The ratio test essentially tells you that the the terms (eventually) act like a geometric series. Therefore, if you can get a handle on how close something is to being a geometric series, you can use the geometric series computation to bound the remainder.

  • If you have a recurrence like $x_n=ax_{n-1}+b$, then you can prove that if the sequence converges, then it must converge to a fixed point, i.e., a point that satisfies $L=aL+b$, $L=\frac{b}{1-a}$. However, to use this, one must first justify why the sequence converges at all.

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  • $\begingroup$ :o I didn't know about this "bounds on error" thing... thanks! $\endgroup$ – K Split X Apr 20 '17 at 17:08
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If the series $\sum_{n=0}^\infty a_n$ converges, then I already know what it converges to: the actual value is $\sum_{n=0}^\infty a_n$.

There are lots of ways to represent numeric values, and "the sum of this series" is one of them. Often, it is a very useful way to write the actual value. Sometimes, it's even the best way we have to do so.

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I don't want to replace any of the answers given so far and rather stress just one point: To sometimes consider the questions of convergence independently from the computation of the limit is an important educational step, in order to build up a more abstract point of view on convergence.

  • I guess, there are many reasons to discuss convergence of series as one of the first topics in an analysis course. Analysis of series is a bit more abstract, than analysis of sequences, but simple enough to present elementary proofs for the important theorems. Students see how exchange of limits can lead to wrong results. (This can be also archived with sequences, but harmonic series as counterexample is simpler to remember.) They can to practice how to estimate and handle an abstract term like a sum with unknown entries.

Why is it important to consider convergence and computation of limit points separate ?

  • Some spaces are constructed to be the completion of certain smaller spaces. For example the real numbers can be constructed as the set of all limit points of rational Cauchy-convergent sequences (with identifying some limit points as being equal). Or sometimes you build function spaces in this way, for example the set of limits of smooth functions with respect to some Sobolev norm. When working in these spaces, it is often important to handle convergence of sequences without using a closed form for the limit point.

  • Many numerical simulations use some kind of series or sequences to approximate complicated terms, like solutions of differential equations. In these situations the computer will compute sufficiently many terms to estimate the exact limit. It is of great interest to prove theoretically, that these series converge to a unique limit. In many cases the limit of the sequence exist only as an abstract object, being a solution of a certain differential equation. It is common the consider the general convergence to some point (~ Stability) independently from conditions to ensure that an existing unique limit is indeed the correct solution (~ Consistency). Therefore I consider it to be an important lesson to see how convergence and computing the limit point can be treated somehow independently from each. It can happen surprisingly easy, that numerical simulations converge to a false solutions if one of both condition is not satisfied, see for example the plots one the first slides here: Slides 3-5.

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