Uniform Convergence using Abel's test for a series based on convergence of a series

Question

A problem from uniform convergence of series:$$\sum_{i=1}^\infty a_n$$ is convergent then show that $$\sum_{i=1}^\infty \frac {nx^n(1-x)}{1+x^n} a_n$$ and $$\sum_{i=1}^\infty \frac {2nx^n(1-x)}{1+x^{2n}} a_n$$ are uniformly convergent when $x \in\ [0,1]$.

Answer 1

As per second thoughts @tom-tom, requesting you to look into this

$ 0\le x \le 1 \Rightarrow x^r \le 1 \quad \forall \ r \ge 0 $

$\Rightarrow 0\le x^{n} \le x^r \quad \forall \ r\le n \ \& \ r \ge 0$

$\therefore \sum_{r=0}^{n-1} x^r \ge n.x^n \ge 0$

$ \Rightarrow \frac {1-x^n}{1-x} \ge nx^n \ge 0$

$\Rightarrow 0\le\frac {nx^n(1-x)}{1-x^n} \le 1$

$$\Rightarrow 0\le\frac {nx^n(1-x)}{1+x^n} \le 1$$

$ \ <b_n(x)>\ = \frac {nx^n(1-x)}{1+x^n}$ is bounded.

$\sum a_n$ is given as convergent. As it is free from $x$, $\therefore$ it is uniformly convergent.

Now, $ \ 1. \ <a_n(x)> $ is uniformly convergent, $ \ 2. \ <b_n(x)> $ is bounded for $ x \in [0,1] \ $, $\quad$ 3. $\ <b_n(x)> $ shall be piecewise monotonic.

$\therefore$ $ \quad \sum a_n(x) b_n(x)$ shall be uniformly convergent.

$$ \Rightarrow \sum_{i=1}^\infty \frac {nx^n(1-x)}{1+x^n} a_n \ $$ is convergent.

Similarly other one can be proved. Please let me know of mistakes, if any.