Sequence And Series Question 315
Question: The sum of the series $ \frac{1}{1.2}+\frac{1.3}{1.2.3.4}+\frac{1.3.5}{1.2.3.4.5.6}+…..\infty $ is
[Kurukshetra CEE 2002]
Options:
A) $ 15e $
B) $ {e^{1/2}}+e $
C) $ {e^{1/2}}-1 $
D) $ {e^{1/2}}-e $
Show Answer
Answer:
Correct Answer: C
Solution:
The nth term of given series is $ T_{n}=\frac{1.3.5.7…..(2n-1)}{1.2.3.4………(2n)} $ $ T_{n}=\frac{1.2.3.4…..(2n-2)(2n-1)(2n)}{1.2.3.4…(2n-1)(2n)} $ $ \times \frac{1}{2.4.6…(2n-2)(2n)} $ $ T_{n} $ $ =\frac{1}{(2^{n}n!)} $ , $ \therefore S=\sum\limits_{n=1}^{\infty }{\frac{{{( \frac{1}{2} )}^{n}}}{n!}={e^{\frac{1}{2}}}-1.} $
BETA
