So basically there are a bunch of tests (integral test, ratio test, p-series test, etc) that tells us whether a series converges or not, but none of them gives any information about the limit of a convergent series.
Unless you are lucky and have a convergent geometric series in which case the limit can be found by the formula $\frac{1}{1-r}$.
Or if you are lucky and can somehow formulate the nth partial sum of a series and find the limit as n tends to infinity.
I'm wondering if there is a general method to find the limit of a convergent series?
Best Answer
There's no general method, and when it is possible, it can be very hard.
A good example is the Riemann zeta series: $$\zeta(s)=\sum_{k=1}^\infty\frac1{n^s},$$ which converges for $\mathrm{Re\,(s)>1}$.
For $s$ even integer, its limits have been known since Euler, for instance $$\zeta(2)=\frac{\pi^2}6,\quad\zeta(4)=\frac{\pi^4}{90},\; \&c. $$ However, its values for odd integers are not known (except for numerical approximations), and it was only in 1979 that Roger Apéry proved $\zeta(3)$ is irrational; in 2000, Tanguy Rivoal proved an infinity of $\zeta(s)$ (s odd integer) are irrational, and it is only conjectured they're all irrational.