1

Is the $\sum_{n=1}^\infty \frac{nx}{1+n^3x^2} $ uniformly convergent on $(0,\infty)$?

Each individual term attains a max of $\frac{1}{2n^{1/2}} $ at $x=\frac{1}{n^{1.5}}$.

Notice that the maximizers move to the left and maximums shrink as $n$ increases. So, one cannot put a lower bound of $\sum \frac{1}{2n^{0.5}}$ in a crude way. But I believe with some more algebra we might still bound from below by the latter sum which diverges. And that is what I am asking!

Behnam Esmayli
  • 5,024
  • 2
    Consider $g(x):=x^{-2/3}\in L^1[0,1]$. For your series $\sum_{k\ge1}f_k(x)$, it is easy to check $\int_0^1 f_k g, dx\ge {1\over2k}$. Therefore $\int_0^1\big(\sum_{k\ge1}f_k\big)g, dx=+\infty$. So $\sum_{k\ge1}f_k$ is not bounded on $[0,1]$ and the convergence is not uniform. – Pietro Majer Nov 09 '19 at 14:08

0 Answers0