Convergence and sum of $\sum_{n=0}^\infty \frac{x}{(2nx-x+1)(2nx+x+1)}$

Question

Find the set of $x$ where:

$$\sum_{n=0}^\infty \frac{x}{(2nx-x+1)(2nx+x+1)}$$

converges and calculate the sum. Determine where does the series converge uniformly.

Would appreciate any help,

Answer 1

Hint. One may observe that, by partial fraction decomposition, we have $$ \frac{x}{(2nx-x+1)(2nx+x+1)}=\frac12\frac1{((2n-1)x+1)}-\frac12\frac1{((2n+1)x+1)} $$ then, summing from $n=1$ to $N$, terms telescope giving, for $x\neq1$,

$$ \sum_{n=0}^N\frac{x}{(2nx-x+1)(2nx+x+1)}=\frac12\frac1{1-x}-\frac12\frac1{((2N+1)x+1)} $$

This may be easier for you to conclude.

Answer 2

The specific terms of this sum simplify to $a(n)=\frac{x}{(2nx-x+1)(2nx+x+1)}= \frac{x}{4n^2 x^2 + 4nx-x^2+1}$, and this makes certain tests of convergence easier to calculate.

The ratio test is inconclusive: $\lim_{n \to ∞}\frac{a(n+1)}{a(n)} = \lim_{n \to ∞}\frac{4(n+1)^2 x^2 + 4(n+1)x-x^2+1}{4n^2 x^2 + 4nx-x^2+1} = 1$ Do you know what the other tests are that mathematicians use?