Uniform convergence of $\sum_{n=1}^\infty \frac{1}{n^3+x}$ How do is show that for $-1<x<1$ the series $\sum_{n=1}^\infty \frac{1}{n^3+x}$ converges uniformly.
For $x\geq0$ I can make a convergent Majorant series, choosing $M_n = 1/n^3$, but I can't seem to determine a convergent Majorant series for $-1<x<0$.

Actually, you could use a simple comparison test.  Since $|\frac 1{n^2}|>|\frac 1 {n^3-x}| for -1<x<1$, and since the sum of the reciprocal squares is uniformly convergent, then it follows that your original sum is also convergent.
