TIFR-2016-Maths-A: 3

Question

The value of the series $\sum _{n=1}^{\infty }\frac{n}{2^{n}}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$.

Answer 1

$\sum_{n=1}^{ \infty}$ $n/ 2^n$ 

$\Rightarrow $  $Let$  $S$  $=1/2^1 +2/2^2$ $+3/2^3 + ...$   $---- (1)$

      $(1/2 )S =$                 $1/2^2 +2/2^3$ $+3/2^3 + ...$  $----(2)$           $subtract$  $1$  $and $  $2 $

$-------------------------$

  $(1/2 )S =$  =      $1/2^1 +1/2^2 +1/2^3$ $+1/2^3 + ...$  

Sum of infinite series $S_{\infty} = a/1-r$

$1/2S_{\infty}  =$  $1/2 / 1/2 $

$S_{\infty} $=  $2$

$So $  $ B $  $ should $  $ be  $  $ the $  $ ans $

 

Comments

Nice explanation

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@mikasa_ackermann 😜