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julian's latest activity
J
julian
replied to the thread
POTW
Sum of an Alternating Series
.
Hi @anuttarasammyak, I'm looking at your solution method from post #8 in more detail. For simplicity reasons I consider ##e^{inx}##...
Yesterday, 10:01 AM
J
julian
replied to the thread
POTW
Sum of an Alternating Series
.
The correct formula is \begin{align*} \sum_{n=1}^{\infty}\frac{\sin(nx)}{n^3} = - \frac{1}{12} x (x-\pi)(2\pi-x) \end{align*} and it...
May 20, 2024
J
julian
reacted to
mathwonk's post
in the thread
POTW
Sum of an Alternating Series
with
Informative
.
As those know who learned precalculus from Euler, (Intro. to Analysis of the Infinite, paragraphs ##174-175##, one can deduce from his...
May 17, 2024
J
julian
replied to the thread
POTW
Fourier Series on the Unit Interval
.
It is much easier to evaluate ##\sum_{n=1}^\infty \dfrac{\cos (2 \pi n x)}{n^2}## by Taylor expanding ##I(\alpha)## and obtaining the...
May 17, 2024
J
julian
replied to the thread
I
Question on an infinite summation series
.
General sum, its integral representations, and evaluation of integral The general sum, ##\sum_{n=0}^\infty \dfrac{1}{(2n+1)^{2k}}##...
May 14, 2024
J
julian
replied to the thread
I
Question on an infinite summation series
.
@phymath7 You asked if there is any integral representation of the series that could facilitate its evaluation. In my notes, I provided...
May 14, 2024
J
julian
replied to the thread
POTW
Fourier Series on the Unit Interval
.
Using a method that employs ##\frac{1}{n^2} = \int_0^\infty u e^{-nu} du##, I write the sum as an integral. I evaluate the integral two...
May 13, 2024
J
julian
reacted to
Euge's post
in the thread
POTW
Sum of an Alternating Series
with
Like
.
Find, with proof, the sum of the alternating series $$\sum_{n = 0}^\infty \frac{(-1)^n}{(2n+1)^3}$$
May 10, 2024
J
julian
replied to the thread
POTW
Sum of an Alternating Series
.
You mentioned your Insight here. Reading your Insight is where I got the idea to rewrite an alternating sum over odd integers using...
May 10, 2024
J
julian
replied to the thread
POTW
Sum of an Alternating Series
.
This is a different solution method to the ones I've already given. The general sum \begin{align*} \sum_{n=0}^\infty...
May 7, 2024
J
julian
replied to the thread
POTW
Sum of an Alternating Series
.
It so happens that the general sum \begin{align*} \sum_{n=0}^\infty \dfrac{(-1)^n}{(2n+1)^{2k+1}} \end{align*} for ##k=0,1,2,\dots##...
May 7, 2024
J
julian
replied to the thread
POTW
Sum of an Alternating Series
.
@anuttarasammyak. Is the sum ##Re \sum_{n=1}^\infty e^{-i(p-1) \frac{n \pi}{2}}## related to the periodic delta function though? With...
May 1, 2024
J
julian
replied to the thread
POTW
Does the Taylor series for arctan converge at x = 1?
.
Same method I used here: POTW can be used here as well:
May 1, 2024
J
julian
replied to the thread
POTW
Sum of an Alternating Series
.
May 1, 2024
J
julian
replied to the thread
POTW
Sum of an Alternating Series
.
Apr 29, 2024
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