How to find partial sums of infinite series?

Hint:An infinite series is an expression of the form: \[\mathop \sum \limits_{k = 1}^\infty {a_k} = {a_1} + {a_2} + {a_3} + {a_4} + \cdots \].

The infinite sequence\[({a_k})\], also written as \[\{ {a_k}\} \]or \[{a_1},{a_2},{a_3},\]…., is termed the sequence of terms of the infinite series. Going to basics, recall what a partial sum of an infinite series is. And then proceed with the answer.

Complete step by step solution:
If, for every positive integer n, we let \[{s_n}\] be defined as the finite sum,\[{s_n} = \mathop \sum
\limits_{k = 1}^n {a_k} = {a_1} + {a_2} + {a_3} + \cdots + {a_n}\],

then the sequence $({s_n})$,is named as the “sequence of partial sums” of the infinite series.

The two sequences \[({a_k})\] and $({s_n})$ are associated with one another, but they're different, and it is vital to grasp the relationships and differences.

It is the sequence$({s_n})$ that determines whether the first series \[\mathop \sum \limits_{k = 1}^\infty {a_k}\]converges or not ("sums" to a finite number or not). In fact, by definition, if $({s_n})$ converges to a number s, then the series \[\mathop \sum \limits_{k = 1}^\infty {a_k}\]is

also said to converge to s and then we may write that as \[\mathop \sum \limits_{k = 1}^\infty {a_k} = s\]

On the opposite hand, if s diverges, then the series \[\mathop \sum \limits_{k = 1}^\infty
{a_k}\]will also diverge.

Note: If the sequence\[({a_k})\] doesn't converge to zero (meaning it either converges to another number or it diverges), then the sequence$({s_n})$ diverges, meaning that the series \[\mathop \sum \limits_{k = 1}^\infty {a_k}\]diverges.

 If the sequence \[({a_k})\]does converge to zero, then the sequence $({s_n})$may not converge (so the series \[\mathop \sum \limits_{k = 1}^\infty {a_k}\]may not converge).

In such cases, we want other tests to assist us decide about convergence (a formula for the partial sums, ratio test, root test, comparison test, limit comparison test, alternating series test, absolute convergence test, etc.)