Question

Explain what it means to say that\(\sum\limits_{n = 1}^\infty {{a_n}} = 5\).

Short answer

It means that series converges to\(5\).

Step-by-step solution

Definition

A convergent series is an infinite sum of a sequence of numbers\(\sum\limits_{n = 1}^\infty {{a_n}} \)that adds up to finite number.

Explanation

Given a series \(\sum\limits_{n = 1}^\infty {{a_n}} = 5\)

It means that the sum of the infinite terms of the sequence \(\left\{ {{a_n}} \right\}\)is equal to 5.

i.e \({a_1} + {a_2} + {a_3} + \cdots \cdots = 5\)

We know that\(\mathop {\lim }\limits_{k \to \infty } {S_k} = \sum\limits_{n = 1}^\infty {{a_n}} \)where\({s_k}\)denotes the \({k^{{\rm{th }}}}\)partial sum of series.

Then, \(\mathop {\lim }\limits_{k \to \infty } {s_k} = 5\)and says that the series\(\sum\limits_{n = 1}^\infty {{a_n}} \)converges to 5.

And 5 is called the sum of the series\(\sum\limits_{n = 1}^\infty {{a_n}} \).