A geometric series has first term and common ratio

MATHEMATICS · Multi-part · CBSE · Mathematics · Sequences and Series · Sequences and Series

Problem

A geometric series has first term

a

and common ratio

r

. Suppose that the series is written as

\displaystyle \sum_{k=0}^{\infty} ar^{k}

. (a) State the condition on

r

that guarantees the existence of the sum of the series. Explain why this condition is necessary. (b) Under the condition you identified in part (a), derive a formula in terms of

a

and

r

for the sum of the series. (c) Evaluate the sum when

a=7

and

r=\dfrac{2}{3}

, showing each step clearly.

Related theory

Geometric Progressions
Infinite Series
Sequences and series
Algebra

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