In the following excercise I have to tell whether the function-series is a) pointwise convergent b) uniform convergent and/or c) compact convergent on intervall $[a, \infty]$ (a > 0).
The series is:
$$\sum_{n=1}^\infty x \cdot [ n^2\cdot e^{-(n+1)x} - (n-1)^2\cdot e^{-nx}].$$
The problem is: I really don't know what I have to do. So I'll be glad for any kind of help :)
a) The series is pointwise convergent, if the function-sequence is pointwise convergent. If my series is pointwise convergent and therefore the sequence as well, I should be able to find the "pointwise limit of the sequence" and this means I have to find a function $f = \lim f_n$. Is this correct so far?
b) The series is uniform convergent, if the function-sequence is uniform convergent. In this case, the sequence $(f_n)$ converges to a function $f$, if $ \forall \varepsilon > 0 \exists n_0 \in \mathbb{N} \forall n \ge n_0 \forall x \in M: |f_n(x) - f(x)| < \varepsilon$.?
c) ?