Sequence And Series Question 453
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}}+….. $ differs from $ S_{n} $ by a quantity $ <{10^{-6}} $ , is
Options:
A) 21
B) 20
C) 19
D) None
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ 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}} $
$ \therefore ,\frac{1}{{2^{n-1}}}<{10^{-6}}\Rightarrow {2^{n-1}}>10^{6} $
$ \therefore ,n-1>6{\log_2}10=\frac{6}{0.3010}n>20 $ $ [ \because ,\frac{6}{.3018}<20 ] $
$ \therefore ,n=21 $
BETA
