Question

help me please...
does the series converges? if it does , try to find the sum.
summation n=1 to infinity n/(2^n)

how should i start solving this question ????

Answer 1

The ratio test shows that this is convergent:\[ \frac{a_{n+1}}{a_{n}}=\frac{\frac{n+1}{2^n}}{\frac{n}{2^n}}=\frac{n+1}{2n}=\frac{1+\frac{1}{n}}{2}\]Taking the limit as n approaches infinity shows that the ratio is 1/2, which is less than 1 (which is what is needed if the ratio test is to work).

Answer 2

thank you