query: how can I calculate the sum of the n first elements : $ 1\cdot 2 +2\cdot2^2 +3\cdot2^3+4\cdot2^4+\cdots+k\cdot2^k+\cdots?$
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negative_passages.text: Sum of (example: 0+1+2+3 = 6 , 0+1+2+3+4+5+6+7 = 28 and so on)
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negative_passages.text: Prove by induction that $\sum_{i=1}^{n} i \cdot 2^i = (n-1) \cdot 2^{n+1} +2$. Help finding my mistake
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negative_passages.text: Sum of the series $\frac{1}{2\cdot 4}+\frac{1\cdot3}{2\cdot4\cdot6}+\dots$
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negative_passages.text: A closed form for the sum $S = \frac {2}{3+1} + \frac {2^2}{3^2+1} + \cdots + \frac {2^{n+1}}{3^{2^n}+1}$ is ...
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negative_passages.text: Finding a formula to sum natural numbers up to $n$
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negative_passages.text: How to calculate the following sum:
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negative_passages.text: Showing that if $p_1 + \cdots p_n = 1$ then $\displaystyle \sum_{k=1}^n \left(p_k + \dfrac {1}{p_k} \right)^2 \ge n^3+2n+\dfrac 1n$?
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negative_passages.text: How to derive the formula for the sum of the first $n$ odd numbers: $n^2=\sum_{k=1}^n(2k-1).$