An Improper Riemann Integral
Inviato:
12/03/2015, 21:53da gugo82
Exercise
Evaluate:
\[
\intop_0^\infty \frac{x^3}{e^x - 1}\text{d} x\; .
\]
Inviato:
13/03/2015, 12:51da j18eos
For clarity, one has:
\[
\frac{e^x}{1-e^x}=\sum_{n=1}^{+\infty}e^{nx}\,\text{for}\,x<0;
\]
formally, one has:
\[
\frac{e^{-x}}{1-e^{-x}}=\sum_{n=1}^{+\infty}\left(e^{-x}\right)^n\,\text{for}\,x>0.
\]
I think that you are wrong...
Re: An Improper Riemann Integral
Inviato:
13/03/2015, 13:50da gugo82
@ TeM: Ok, your answer's correct!
***
Now a simple generalization...
Exercise
Evaluate:
\[
\intop_0^\infty \frac{x^\alpha}{e^x - 1}\ \text{d} x\; ,
\]
where \(\alpha \geq 1\).
Re: An Improper Riemann Integral
Inviato:
13/03/2015, 16:43da gugo82
@ TeM: Abramowitz & Stegun?
Re: An Improper Riemann Integral
Inviato:
13/03/2015, 21:23da gugo82
TeM ha scritto:gugo82 ha scritto:Abramowitz & Stegun?
Honestly I didn't know anything of this "Bible" (I peeked just now on the net...)!!
It is a very useful reference for this kind of questions.
There is also a
new online edition.
TeM ha scritto:The idea for the series came to me reviewing some similar example that our professor of
analysis mathematics showed us in his "insights for the most curious", while for all the rest
(as was ever mine wont) the source is
mathworld.wolfram.com.
Just out of curiosity, who was your Mathematical Analysis prof.?