All Questions
4
questions
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votes
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answers
59
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Interchange of differentiation and summation in infinite sums
Is it possible to interchange differentiation and summation for infinite but also uniformly convergent sums, like:
$\dfrac{d}{dx} \sin{x} = \dfrac{d}{dx}\sum_{n=0}^\infty (-1)^{n} \, \dfrac{x^{2n + 1}}...
- 181
0
votes
0
answers
33
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Convergence of specific power series
I have to evaluate pointwise/uniform/total convergence of this series and I didn't quite understand how to do it.
$$\sum_{k=2}^{+\infty}{\ln k \over 2+\sin k}x^k$$
For pointwise convergence: it ...
- 415
1
vote
2
answers
89
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Study simple convergence of $\sum_{n=0}^{\infty} x^{2n}$ on $[0,1[$
I have to study
1 ) the simple convergence of
$$S(x) = \sum_{n=0}^{\infty} x^{2n}$$
and
2) the uniform convergence
My attempts :
1)
$\forall x \in [0,1[$ $$S_n(x) = \frac{1-(x^2)^{n+1}}{1-x^2}...
- 239
0
votes
1
answer
1k
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How to find sum of the power series $\sum_{n=1}^{\infty} x^2 e^{-nx}$?
I got this power series $$\sum_{n=1}^{\infty} x^2 e^{-nx} $$
And i need to prove uniform convergence when $x \in[0;1]$ and find this sum. I have proven uniform convergence, but i have no idea how to ...
- 535