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Let $\sum\limits_{n = 0}^{+ \infty} u_{n}$ be a convergent series, such that $\forall n , u_{n} > 0$

My question is : Under which conditions can we find a constant $C > 0$ such that

$$\forall m ,\;\;\sum\limits_{n = m}^{+ \infty} u_{n} \leq C \, u_{m} $$

For example, if $u_{n} = \frac{1}{n!}$, then $\sum\limits_{n = m}^{+ \infty} u_{n} = \frac{e}{m!} \leq C \frac{1}{m!}$

However, for $u_{n} = \frac{1}{n^{2}}$, we have $\sum\limits_{n = m}^{+ \infty} u_{n} = PolyGamma[1,m]$, and $\lim\limits_{m \to + \infty}PolyGamma[1,m] \, m^{2} = + \infty $...

Any sufficient or necessary conditions would be great !

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