Discussion Forum



1)

Assertion (A)    if |x|<1, then   

$\sum_{m=0}^{\infty}(-1)^{n} x^{n+1}=\frac{x}{x+1}$

Reason  (R)    if |x|<1, then $(1+x)^{-1}$ = $ 1-x+x^{2}-x^{3}$+.....

Which one of the following is true?


A) (A) and (R) are true , (R) is correct explanation of (A)

B) (A) and (R) are true but (R) is not a correct explanation of (A)

C) (A) is true , but (R) is false

D) (A) is false , but (R) is true

Answer:

Option A

Explanation:

 We have,

 $\frac{x}{x+1}= x(1+x)^{-1}=x(1-x+x^{2}-x^{3}+x^{4}-...)$

  = $x-x^{2}+x^{3}-x^{4}+x^{5}-$

  $= \sum_{n=0}^{\infty}(-1)^{n} x^{n+1}$

$\therefore$   Assertion  is true

Reason  (R)=$(1+x)^{-1}$ = $ 1-x+x^{2}-x^{3}$+.....

is also true , (A)  and (R)  are true , (R) is correct explanation of (A)