- #1
Hertz
- 180
- 8
Homework Statement
I'm trying to test whether the sequence converges or not:
\sum^{∞}_{k = 1}ke^{-2k^2}
2. The attempt at a solution
I tried to evaluate this in two ways, each of which produced different answers. I was able to eventually discover that this series does converge, but I still don't see what was wrong with the first method I tried (which told me it diverged.)
Could someone please take a look at my work and tell me what I did wrong?
\sum^{∞}_{k = 1}ke^{-2k^2}
\int{^{∞}_{1}xe^{-2x^2} dx}
Let u = -2x^2
du = -4x dx
\frac{-1}{4}\int{^{∞}_{1}-4xe^{-2x^2} dx}
\frac{-1}{4}\int{^{∞}_{-2}e^{u} du}
\frac{-1}{4}{lim}_{b → ∞}[e^u]^{b}_{-2}
\frac{-1}{4}[{lim}_{b → ∞}(e^b) - \frac{1}{e^{2}}]
\frac{-1}{4}[∞ - \frac{1}{e^{2}}]
= -∞
However, if you instead let u = 2x^{2} it can be shown that the series converges. (Along with the integral)
