$$\sum_{n=1}^{ M}{(1/(n*(n+1)*2^(n+1)))}$$

asked by guest
on Jul 06, 2024 at 7:03 am



You asked:

Evaluate the expression: \(\sum_{n = 1}^{M} {\frac{1}{n \cdot \left(n + 1\right) \cdot {2}^{n + 1}}}\)

MathBot Answer:

Evaluated



\(\displaystyle \sum_{n = 1}^{M} {\frac{1}{n \cdot \left(n + 1\right) \cdot {2}^{n + 1}}} = - \frac{2^{- M - 1} \left(\left(- M^{2} - 3 M - 2\right) \Phi\left(\frac{1}{2}, 1, M + 2\right) + \left(M^{2} + 3 M + 2\right) \Phi\left(\frac{1}{2}, 1, M + 1\right)\right)}{2 \left(M + 1\right) \left(M + 2\right)} - \frac{\log{\left(2 \right)}}{2} + \frac{1}{2} \)

Expanded

\[\sum_{n = 1}^{M} {\frac{1}{n \cdot \left(n + 1\right) \cdot {2}^{n + 1}}} = \sum_{n=1}^{M} \frac{1}{2 \cdot 2^{n} n^{2} + 2 \cdot 2^{n} n}\]

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