Sequence And Series Question 1
Question: If $ |x|,<1 $ , then the sum of the series $ 1+2x+3x^{2}+4x^{3}+………..\infty $ will be
Options:
A) $ \frac{1}{1-x} $
B) $ \frac{1}{1+x} $
C) $ \frac{1}{{{(1+x)}^{2}}} $
D) $ \frac{1}{{{(1-x)}^{2}}} $
Show Answer
Answer:
Correct Answer: D
Solution:
This is an A.G.P. Let $ S=1+2x+3x^{2}+…….\infty $
$ \Rightarrow $ $ x.S=x+2x^{2}+……..\infty $ Subtracting $ (1-x)S=1+x+x^{2}+………\infty =\frac{1}{1-x} $
$ \therefore $ $ S=\frac{1}{{{(1-x)}^{2}}} $ . Aliter : Use $ S=[ 1+\frac{r}{1-r}\times diff\text{.}\ of\ A\text{.P}\text{.} ]\frac{1}{1-r} $ .
BETA
