The least value of n a natural number- for which the sum S of the series 1- 1-2-1-22-1-23- - differs from Sn by a quantity -10-6 is

S= 1/1 1/2=2 S_n= 1(1 1/2^n )/1 1/2= 2 1/2^n 1 Given S S_n lt;10^ 6 ∴1/2^n 1 lt;10^ 6⇒ 2^n 1 gt;10^6 ∴ n 1 gt; 6 log_2 10 = 6/0.3010⇒ n gt;20 [∵6/.30 ...

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Standard XIII

Mathematics

Infinite GP

The least val...

Question

The least value of n ( a natural number), for which the sum S of the series $1 + \frac{1}{2} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + \dots$ differs from $S_{n}$ by a quantity $< 10^{- 6}$ is

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Solution

The correct option is A 21
$S = \frac{1}{1 - \frac{1}{2}} = 2 S_{n} = \frac{1 (1 - \frac{1}{2^{n}})}{1 - \frac{1}{2}} = 2 - \frac{1}{2^{n - 1}}$
Given $S - S_{n} < 10^{- 6} ∴ \frac{1}{2^{n - 1}} < 10^{- 6} \Rightarrow 2^{n - 1} > 10^{6} ∴ n - 1 > 6 l o g_{2} 10 = \frac{6}{0.3010} \Rightarrow n > 20 [∵ \frac{6}{.3018} < 20] ∴ n = 21$

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Similar questions

Q. The least value of n ( a natural number), for which the sum S of the series

1 + \frac{1}{2} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + \dots

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< 10^{- 6}

Q. If S denotes the sum to infinity and

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denotes the sum to infinity and

S_{n}

the sum of

n

terms of the series

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The least value of n ( a natural number), for which the sum S of the series 1+12+122+123+⋯ differs from Sn by a quantity <10−6 is

The correct option is A 21S=11−12=2Sn=1(1−12n)1−12=2−12n−1 Given S−Sn<10−6∴12n−1<10−6⇒2n−1>106∴n−1>6log210=60.3010⇒n>20[∵6.3018<20]∴n=21

The least value of n ( a natural number), for which the sum S of the series $1 + \frac{1}{2} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + \dots$ differs from $S_{n}$ by a quantity $< 10^{- 6}$ is