For some $k < \infty$, and $p \in (0,1)$, consider the following sum: $S_k = \sum_{n=1}^{k} \frac{1}{n^p}$.
What is a closed form expression for $S_k$?
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It may worth have a look at Hurwitz's zeta function. – Kemono Chen Nov 16 '18 at 06:12
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@KemonoChen: Hurwitz's zeta function is also infinite, check generalized harmonic numbers. – Tianlalu Nov 16 '18 at 06:17
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@Tianlalu You can make a subtraction from Riemann's zeta function. – Kemono Chen Nov 16 '18 at 06:20
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Thanks guys. I found the answer below helpful: https://math.stackexchange.com/questions/451558/how-to-find-the-sum-of-this-series-1-frac12-frac13-frac14-do – math_phile Nov 16 '18 at 06:45
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I am not aware of any closed form, but there are some good approximations. The easiest way would be to approximate this sum with an integral: $$\sum_{n=1}^kn^{-p}\approx1+\int_1^k x^{-p}dx=1+\left.\frac1{1-p}x^{1-p}\right\rvert^k_0=1+\frac{k^{1-p}}{1-p}.$$
YiFan Tey
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