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### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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

It means that series converges to$$5$$.

See the 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$$\sum\limits_{n = 1}^\infty {{a_n}}$$.