All Questions
6
questions
2
votes
1
answer
189
views
Evaluating $\sum_{n=1}^\infty \frac{\sin(nx)}{n}$ without integrating $\sum_{n=1}^\infty e^{nx}$
I am looking for alternative solutions for finding this sum
$$\sum_{n=1}^\infty \frac{\sin(nx)}{n} $$
My solution proceeds by integrating $$\sum_{n=1}^\infty e^{nx}=\frac{e^{ix}}{1-e^{ix}}$$
With ...
- 313
1
vote
2
answers
112
views
$\sin(ωt) + \sin(ωt+\Delta\theta) + \sin(ωt+2\,\Delta\theta) + \cdots + \sin(ωt+\overline{n-1}\,\Delta\theta) ={}$? [duplicate]
How will we add this series?
Any simple process?
This is one of the equation derived from the equation to find intesity of ray at any particular point in Fraunhofer diffraction.
I think this can be ...
3
votes
3
answers
182
views
Calculate $\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$
How can we evaluate the following integral:
$$\int_0^{1/10}\sum_{k=0}^9 \frac{1}{\sqrt{1+(x+\frac{k}{10})^2}}dx$$
I know basically how to calculate by using the substitution $x=\tan{\theta}$ :
...
- 109
3
votes
1
answer
71
views
Trigonometric Integration + Series
I am doing an integration question:
$$\int \frac{1-\cos^{2m}x}{1-\cos^2x}$$
So I have to show that $$\frac{1-\cos^{2m}x}{1-\cos^2x}=1+\cos^2x+\cos^4x+...+\cos^{2(m-1)}$$
How can I do that?
- 1,542
4
votes
3
answers
194
views
Help in manipulating Integrals
I try to express : $\displaystyle 1+2\sum _{ k=1 }^n \cos(2k\theta ) $
as : $\dfrac { \sin\left( \theta +2\theta n \right) }{ \sin\left( \theta \right) } $
I tried to use the exponential function :...
- 153
1
vote
1
answer
415
views
Expressing $\int \tan^n x\,dx$ with a sum
I was playing around with integrals of $\tan x$, because I knew that both $\int\tan x\,dx$ and $\int\tan^2x\,dx$ were solvable. I then came across the fact that
$$\begin{align}
\int \tan^n x\,dx &...
- 1,535