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Sequence and Series

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Question : 25 of 86

Marks: +1, -0

The sum of the series

1+10^{-1}+10^{-2}+10^{-3}\dots

\infty

is

$\;\frac{9}{10}$
$\;\frac{1}{10}$
$\;\frac{10}{9}$
None

Solution:

Concept:

Infinite geometric series sum formula:

S_{\infty} = \frac{a}{1-r}

for

|r|<1

.

Explanation:

The series is

1 + 10^{-1} + 10^{-2} + 10^{-3} + \cdots

which is geometric with first term

a = 1

and common ratio

r = 10^{-1} = \frac{1}{10}

.

Since

|r| = \frac{1}{10} < 1

, the sum exists.

Using the formula:

S_{\infty} = \frac{1}{1 - \frac{1}{10}} = \frac{1}{\frac{9}{10}} = \frac{10}{9}

.

Answer:

\frac{10}{9}

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