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Q.
The value of the sum $\displaystyle\sum_{ k =1}^{\infty} \displaystyle\sum_{ n =1}^{\infty} \frac{ k }{2^{ n + k }}$ is equal to
Sequences and Series
Solution:
$\displaystyle\sum_{ k =1}^{\infty} \frac{ k }{2^{ k }} \displaystyle\sum_{ n =1}^{\infty} \frac{1}{2^{ n }}=\displaystyle\sum_{ k =1}^{\infty} \frac{ k }{2^{ k }}\left[\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\ldots \ldots . . . \infty\right]=\displaystyle\sum_{ k =1}^{\infty} \frac{ k }{2^{ k }}=2$