# Uniform convergence of $\sum\limits_{n=1}^{\infty}\frac{nx-1}{n^2}$ for $x\in[2,5]$

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?

• When $x=2$, we get the series $\sum_{n=1}^{\infty}\frac{2n-1}{n^2}$. This series is divergent. – User3101 Dec 7 '15 at 15:08
• Big hint: Limit test for convergence of sums. – zickens Dec 7 '15 at 15:10

Since $\frac{nx-1}{n^{2}}\geq\frac{nx-n}{n^{2}}=\frac{\left(x-1\right)}{n}\geq\frac{1}{n}$,