Find the convergence area of $\sum^\infty_{n=2}((n+1)\ln(\frac{n^2+x^2}{n^2})-n\ln(\frac{(n-1)^2+x^2}{(n-1)^2}))$

Question

I need to find the convergence area of $\sum^\infty_{n=2}((n+1)\ln(\frac{n^2+x^2}{n^2})-n\ln(\frac{(n-1)^2+x^2}{(n-1)^2}))$ and see if it's uniformly convergence in this area.

I can see that every partial sum looks like this:

$S_k(x)=(k+1)\ln(1+\frac{x^2}{k^2})-2\ln(1+x^2)$

and therefore:

$S(x) = \lim_{k\to\infty}S_k(x) = -2\ln(1+x^2)$.

Can I tell the convergence or uniformly convergence from this infromation? If so, how, and why?

Answer 1

Your $S_k (x) $ is correct. now use the equivalence

$$\ln (1+X)\sim X \;\;(X\to 0) $$ or

$$\ln (1+x^2/k^2)\sim x^2/k^2 \;(k\to +\infty) $$

to get $$\lim_{k\to +\infty}S_k (x)=-2\ln (1+x^2) $$ this proves, the series converges (pointwise) for each $x\in\mathbb R $.

the sum function is $$S:x\mapsto -2\ln(1+x^2). $$

For uniform convergence, find the maximum of $$F_n (x)=|(n+1)\ln (1+x^2/n^2)|$$ .

observe that $F_n (n)=(n+1)\ln (2) $ and conclude.