# Find out if sum_(k=1)^oo((k+1)/(2^k)) converges

Find out if $\sum _{k=1}^{\mathrm{\infty }}\frac{k+1}{{2}^{k}}$ converges
I have split it into
$\sum _{k=1}^{\mathrm{\infty }}\frac{k+1}{{2}^{k}}=\sum _{k=1}^{\mathrm{\infty }}\frac{k}{{2}^{k}}+\sum _{k=1}^{\mathrm{\infty }}\frac{1}{{2}^{k}},$
and applied the geometric series to the second part of the sum. But how do I deal with the first one to find the limit?
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Dillon Levy
Because for all $k\ge 10$ we have ${2}^{k}>{k}^{3}$ and
$\sum _{k=10}^{+\mathrm{\infty }}\frac{1}{{k}^{2}}$
converges.