First let’s look if the series converges absolutely.

For this, we need to see if \(\displaystyle\sum{b}_{{n}}=\sum{\frac{{{n}}}{{{2}^{{n}}}}}\) converges. And this is immediate using the ratio test

as \(\displaystyle\lim_{{{n}\to\infty}}{\frac{{{b}_{{{n}+{1}}}}}{{{b}_{{n}}}}}=\frac{{1}}{{2}}{ < }{1}\)

Conclusion: the given series converges absolutely hence converges

For this, we need to see if \(\displaystyle\sum{b}_{{n}}=\sum{\frac{{{n}}}{{{2}^{{n}}}}}\) converges. And this is immediate using the ratio test

as \(\displaystyle\lim_{{{n}\to\infty}}{\frac{{{b}_{{{n}+{1}}}}}{{{b}_{{n}}}}}=\frac{{1}}{{2}}{ < }{1}\)

Conclusion: the given series converges absolutely hence converges