I'm really stuck with my homework in real analysis. Could anyone give me some ideas/tips or solutions, how to get these following tasks done? I would be very thankful!
We have a series
$$\sum_{k=2}^{\infty} x^k(1-x) $$
The following tasks are need to be done for this series:
1) Find the sum of the series $S(x)$.
2) Does this series uniformly converge to a sum $S(x)$ in $[0,1)$.
Consider the partial sums.
$\begin{array}\\ s_n(x) &=\sum_{k=2}^{n} x^k(1-x)\\ &=\sum_{k=2}^{n} (x^k-x^{k+1})\\ &=\sum_{k=2}^{n} x^k-\sum_{k=2}^nx^{k+1}\\ &=\sum_{k=2}^{n} x^k-\sum_{k=3}^{n+1}x^{k}\\ &=x^2-x^{n+1}\\ \end{array} $
These converge if $-1 < x \le 1$ and diverge otherwise.
If $-1 < x < 1$ the sum is $x^2$. If $x = 1$ the sum is $0$ (which is obvious from the definition since each term is then $0$).