# How do I find the limit of a series?

There is no general method to do this.

Consider the following definition considering a series of terms ${a}_{k}$:

${s}_{n} = {\sum}_{k = 0}^{n} {a}_{k}$

This is clearly a sequence, called the sequence of partial sums of the series.

The question of determining the limit of the series

sum_(k=0)^oo a_k = lim_(n to oo) sum_(k=0)^n a_k = ?

then becomes a question of determining the limit of a sequence:

 lim_(n to oo) sum_(k=0)^n a_k = lim_(n to oo) s_n = ?

which does not have a trivial answer.

Some limits can be determined using certain tricks, like the geometric series of initial term $a$ and ratio $r$ such that $- 1 < r < 1$, whose limit can be determined by a simple algebraic trick:

${\lim}_{n \to \infty} {s}_{n} = s = a + a r + a {r}^{2} + \cdots + a {r}^{n} + \cdots$
$r s = a r + a {r}^{2} + a {r}^{3} + \cdots + a {r}^{n} + \cdots$
$s - r s = a \implies s = \frac{a}{1 - r}$

or the the series ${\sum}_{k = 0}^{\infty} \frac{1}{{k}^{2}}$, whose limit can be found using Fourier Series (this question was called the Basel Problem).