Sequence And Series Question 511
Question: The sum of the series $ 1+\frac{1.3}{6}+\frac{1.3.5}{6.8}+….\infty $ is
[UPSEAT 2001]
Options:
A) 1
B) 0
C) $ \infty $
D) 4
Show Answer
Answer:
Correct Answer: D
Solution:
Let, $ S=1+\frac{1.3}{6}+\frac{1.3.5}{6.8}+…\infty $
Þ $ \frac{S}{4}=\frac{1}{4}+\frac{1.3}{4.6}+\frac{1.3.5}{4.6.8}+….\infty $
Þ $ \frac{1}{2}-\frac{S}{8}=\frac{1}{2}-\frac{1}{2}.\frac{1}{4}-\frac{1}{2}.\frac{1.3}{4.6}-\frac{1}{2}\frac{1.3.5}{4.6.8}-….\infty $
Þ $ \frac{1}{2}-\frac{S}{8}=1-\frac{1}{2}+\frac{\frac{1}{2}( \frac{1}{2}-1 )}{1.2}- $ $ \frac{\frac{1}{2}( \frac{1}{2}-1 ),( \frac{1}{2}-2 )}{1.2.3} $ $ +\frac{\frac{1}{2}( \frac{1}{2}-1 )( \frac{1}{2}-2 )( \frac{1}{2}-3 )}{1.2.3.4}….\infty $
Þ $ 1/2-S/8={{(1-1)}^{1/2}}=0 $
Þ $ S/8=1/2\Rightarrow S=4 $ .
BETA
