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Does this series converge uniformly or not?

$$\sum\limits_{n=1}^{\infty}\frac{nx-1}{n^2},\space x\in [2,5]$$

I know the Weierstrass M-test can be used to show if a series converges uniformly, but how would I show that my series does not converge uniformly?

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  • $\begingroup$ When $x=2$, we get the series $\sum_{n=1}^{\infty}\frac{2n-1}{n^2}$. This series is divergent. $\endgroup$ – User3101 Dec 7 '15 at 15:08
  • $\begingroup$ Big hint: Limit test for convergence of sums. $\endgroup$ – zickens Dec 7 '15 at 15:10
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The series even doesn't converge.
Since $\frac{nx-1}{n^{2}}\geq\frac{nx-n}{n^{2}}=\frac{\left(x-1\right)}{n}\geq\frac{1}{n}$,
The series is always greater than or equal to the harmonic series, which diverges.

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