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The value of \frac{\mathrm{C}_0}{1}+\frac{\mathrm{C}_2}{3}+\frac{\mathrm{C}_4}{5}+\ldots is

Option: 1

\frac{2^{\mathrm{n}}}{\mathrm{n}+1}


Option: 2

\frac{2^{\mathrm{n}}-1}{\mathrm{n}+1}


Option: 3

\frac{2^{\mathrm{n}+1}}{\mathrm{n}+1}


Option: 4

none of these.


Answers (1)

best_answer

We have
\begin{aligned} & (1+x)^n=\sum_{r=0}^n C_r x^r \dots..(i)\\ & (1-x)^n=\sum_{r=0}^n(-1)^r C_r x^r \dots...(ii)\end{aligned}

Adding equations (i) and (ii) we have,

(1+x)^n+(1-x)^n=C_0+C_2 x^2+C_4 x^4+\ldots

Integrating both sides with respect to x from 0 to 1, we get

\frac{2^{\mathrm{n}+1}}{\mathrm{n}+1}=\frac{\mathrm{C}_0}{1}+\frac{\mathrm{C}_2}{3}+\frac{\mathrm{C}_4}{5}+\ldots
 

Posted by

Ritika Kankaria

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