Let $k \ge 2$ be an ineteger. Consider the series $$ \sum_{n=1}^{\infty} \frac{1}{n^k} $$ this is typically denoted $\zeta(k)$ as it is the value of the Riemann zeta function at $k$.

Now, Euler showed that for even $k$ this is equl to $$q_k \pi^k $$ where $q_k$ is some rational number (that one can also describe explicty).

This is on the one hand an interesting fact and would also allow to calculate approximations of the powers of $\pi$ but what I actually want to get at is that from this it follows that if one knows that $\pi$ is transcendental then one gets directly that $\zeta(k)$ is transcendental and in particular not a rational number.

So this is for even $k$. What about odd $k$? Say $k=3$. Is this rational or irrational? This question was open for a long time until it was proved at the end of the 1970s by Apéry.

How does this prove go (very roughly!):

He first showed that $$ \zeta(3) = \frac{5}{2} \sum_{i=1}^{\infty} \frac{ (-1)^n}{n^3 C (2n , n )} $$ where $C(2n,n)$ is just the obvious binomial coefficient, which I fail to be able to type properly.

So one could say he evaluated the series on the right in 'a closed form'; showing that its values is something already known/defined.

Then based on this he derived some sequences of rational numbers that converge to $\zeta(3)$ so fast that it is impossible for $\zeta(3)$ to be rational itself thus proving the irrationality of $\zeta(3)$.

So, in order to show that $\zeta(3)$ is irrational he first needed to show that it is equal to the limit of this (other) series, or put differently to evaluate this series; not only the first simpler one.

It would be interesting to be able to do something like this for other odd numbers, but so far noone knows how to do so and the irrationality of $\zeta$ at any other odd positive integers is unknown (although there are results that assert that among certain collections of them there are at least some irrational ones).

Thus finding an evaluation of a series can be used to infer something theoretical on its value.

This is not always the motivation, but sometimes it is the case that the point is not so much to know the value of the series in order to replace it in some compuation say, but rather to use the series as a form of describing its value by simpler buildingblocks and thereby allowing to learn something (new) on the value.

Let $k \ge 2$ be an integer. Consider the series $$ \sum_{n=1}^{\infty} \frac{1}{n^k} $$ this is typically denoted $\zeta(k)$ as it is the value of the Riemann zeta function at $k$.

Now, Euler showed that for even $k$ this is equal to $$q_k \pi^k $$ where $q_k$ is some rational number (that one can also describe explictly).

This is on the one hand an interesting fact and would also allow to calculate approximations of the powers of $\pi$ but what I actually want to get at is that from this it follows that if one knows that $\pi$ is transcendental then one gets directly that $\zeta(k)$ is transcendental and in particular not a rational number.

So this is for even $k$. What about odd $k$? Say $k=3$. Is this rational or irrational? This question was open for a long time until it was proved at the end of the 1970s by Apéry.

How does this prove go (very roughly!):

He first showed that $$ \zeta(3) = \frac{5}{2} \sum_{i=1}^{\infty} \frac{ (-1)^n}{n^3 \binom{2n}{n}}.$$

So one could say he evaluated the series on the right in 'a closed form'; showing that its values is something already known/defined.

Then based on this he derived some sequences of rational numbers that converge to $\zeta(3)$ so fast that it is impossible for $\zeta(3)$ to be rational itself thus proving the irrationality of $\zeta(3)$.

So, in order to show that $\zeta(3)$ is irrational he first needed to show that it is equal to the limit of this (other) series, or put differently to evaluate this series; not only the first simpler one.

It would be interesting to be able to do something like this for other odd numbers, but so far no-one knows how to do so and the irrationality of $\zeta$ at any other odd positive integers is unknown (although there are results that assert that among certain collections of them there are at least some irrational ones).

Thus finding an evaluation of a series can be used to infer something theoretical on its value.

This is not always the motivation, but sometimes it is the case that the point is not so much to know the value of the series in order to replace it in some computation say, but rather to use the series as a form of describing its value by simpler building blocks and thereby allowing to learn something (new) on the value.