Answer
First five terms are: $0.5, 0.5,0.375, 0.25, 0.15625$
Converges to $0$.
Work Step by Step
Plugging $n = {1,2,3,4,5}$ in $\dfrac{n}{2^n}$.
$\implies n=1$: $\dfrac{1}{2^1} =0.5$
$\implies n=2$: $\dfrac{2}{2^2} =0.5$
$\implies n=3$: $\dfrac{3}{2^3} =0.375$
$\implies n=4$: $\dfrac{4}{2^4} =0.25$
$\implies n=5$: $\dfrac{5}{2^5} =0.15625$
We see that $\lim\limits_{n \to \infty} \dfrac{n}{2^n}=\lim\limits_{n \to \infty} \dfrac{1}{2^n \ln (2)}\\=\dfrac{1}{\infty}\\=0$
Therefore, the given series converges to $0$.
