Does there exist a closed form for finite summation of the sequence $\sum^n_{i=1}{e^i/i}$ ?
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Take the derivative of the function $f(x) = \sum_{i = 1}^n \frac{x^i}{i}$ which is easier to compute. |
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$$f(x):=\sum_{k=1}^nx^{k-1}=\sum_{k=0}^{n-1}x^k=\frac{1-x^{n-1}}{1-x}$$ Integrate indefinitely the above: $$\int f(x)\,dx=\sum_{k=1}^n\int x^{k-1}dx=\sum_{k=1}^n\frac{x^k}{k}\ldots$$ |
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Maple writes it as $$\sum_{i=1}^n \frac{{\rm e}^i}{i} = -\Phi \left( {{\rm e}},1,n \right)\; {{\rm e}^{n}}-\ln \left( -{{\rm e}}+1 \right) +{\frac {{{\rm e}^{n}}}{n}}$$ where $\Phi$ is the Lerch Phi function. |
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