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Function sequence $(f_n)$ is defined as $f_n(x) :=\frac{1}{n^2} \sum_{i=1}^n i^x$ for $x \in \mathbb{R}$.

  1. I was wondering how to decide its convergence region? If it were a p-series, then there was some standard result, but it isn't a p-series.
  2. In particular, what is the region where $(f_n)$ converges to $0$?

Thank you!

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up vote 1 down vote accepted

By the Euler-Maclaurin summation formula (or the trapezoidal rule) we have $$\sum_{i=1}^n i^x = \frac{n^{x+1}}{x+1} + \frac{n^x}{2} + \mathcal{O}(n^{x-1}).$$

Thus $f_n \to f$ pointwise, where $f(x) = 0 $ for $ x<1$, $ f(1)= \frac{1}{2} $ and $f(x) = \infty .$

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