Show that each sequence of partial sums and its derivative converges uniformly on their respective intervals.
a) $$ S_n(x)=\sum_{k=1}^n \frac{1}{(1 + nx)^2}, x \in (0,\infty)$$
$$ S_n'(x)=\sum_{k=1}^n \frac{-2}{(1 + nx)^3}, x \in (0,\infty)$$
b)$$S_n(x)=\sum_{k=1}^n e^{-nx}, x \in(0,\infty)$$
$$S_n'(x)=\sum_{k=1}^n -ne^{-nx}, x \in(0,\infty)$$
for part a) I believe I can make use of the following $$\sum_{n=1}^\infty \frac{1}{(1+nx)^2} < \sum_{n=1}^\infty \frac{1}{(nx)^2} = \frac{1}{x^2}\sum_{n=1}^\infty \frac{1}{n^2}$$ so the infinite series converges.I know that I need to use Weierstrass M-Test and the Comparison test to show that the sequences $\{S_n(x)\}$ and $\{S_n'(x)\}$ converge uniformly on $(0,\infty)$
Similarly, for part b) $$\sum_{n=1}^\infty e^{-nx} = \sum_{n=1}^\infty \left(e^{-x}\right)^n$$ which shows the infinite series is a geometric series and converges.
I need help showing that $\{S_n(x)\}$ and $\{S_n'(x)\}$ converge uniformly on $(0,\infty)$ for part a) and b).I think I need to show that it converges uniformly on$[a,\infty)$ for every $a>0$ which would hold for all $x\in(0,\infty)$.But im not sure..help anyone?
You have already received clear comments and answers; so, what I shall add is quite marginal but I thought that it could be of interest to you. $$S_n(x)=\sum_{k=1}^n \frac{1}{(1 + kx)^2}=\frac{\psi ^{(1)}\left(1+\frac{1}{x}\right)-\psi ^{(1)}\left(1+\frac{1}{x}+n\right)}{x^2}$$ $$S_n'(x)=\sum_{k=1}^n \frac{-2}{(1 + kx)^3} =\frac{\psi ^{(2)}\left(1+\frac{1}{x}\right)-\psi ^{(2)}\left(1+\frac{1}{x}+n\right)}{x^3}$$ If $n$ goes to $\infty$, the second term in each numerator disappears.
As already told by André Nicolas, the only problems appear for $x=0$. For the remaining, the convergence is quite clear.