Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
cinar
Group Title
I am looking for solution step by step for this integral, I know what the answer is.
\[\int\limits_{0}^{\infty}\frac{x ^{3}}{e ^{x}1}dx=\frac{\pi ^{4}}{15}\]
 2 years ago
 2 years ago
cinar Group Title
I am looking for solution step by step for this integral, I know what the answer is. \[\int\limits_{0}^{\infty}\frac{x ^{3}}{e ^{x}1}dx=\frac{\pi ^{4}}{15}\]
 2 years ago
 2 years ago

This Question is Closed

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
*bookmark. Going to bed now, but I'll write this out for you tomorrow. Nice question.
 2 years ago

cinar Group TitleBest ResponseYou've already chosen the best response.2
tell me about it, it is really hard..
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Ok... \[ \int_0^\infty \frac{x^3}{e^x  1} dx = \int_0^\infty \frac{x^3e^{x}}{1e^{x}} \] \[ = \int_0^\infty \sum_{n=0}^\infty x^3 e^{(n+1)x} dx \] \[ = \sum_{n=0}^\infty \int_0^\infty x^3 e^{(n+1)x} dx \] Changing variable \( u = (n+1)x \), this is equal to \[ \sum_{n=0}^\infty \int_0^\infty \frac{u^3}{(n+1)^4} e^{u} du \] \[ = \sum_{n=0}^\infty \frac{1}{(n+1)^4} \Gamma(4) \ \hbox{, by definition of the Gamma function }\] \[ = 6 \ \sum_{n=1}^\infty \frac{1}{n^4} \ \ \ \ \ \hbox{, as } \Gamma(4) = 3! = 6 \] \[ = 6 \ \frac{\pi^4}{90} \ = \ \frac{\pi^4}{15} \]
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
We can now generalize this to \[ \int_0^\infty \frac{x^t}{e^x  1} dx = \Gamma(t+1)\zeta(t+1) \]
 2 years ago

cinar Group TitleBest ResponseYou've already chosen the best response.2
actually, my question contains \[\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n ^{s}}\] \[\zeta(4)=1+\frac1{2^{4}}+\frac1{3^{4}}+\frac1{3^{4}}...=\frac {\pi ^{4}}{90}\] how it is solution..
 2 years ago

cinar Group TitleBest ResponseYou've already chosen the best response.2
I did not understand how passed this step \[= \int\limits_0^\infty \sum_{n=0}^\infty x^3 e^{(n+1)x} dx\]
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
\[ \frac{1}{1x} = 1 + x + x^2 + x^3 + ... \] provided x < 1. As for \( x \in (0, \infty) \) we have \( 0 < e^{x} < 1 \), \[ \frac{e^{x}}{1e^{x}} = e^{x} ( 1 + e^{x} + e^{2x} + e^{3x} + ...) \] \[ = e^{x} + e^{2x} + e^{3x} + e^{4x} + ... \]
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
does this make sense now?
 2 years ago

cinar Group TitleBest ResponseYou've already chosen the best response.2
yes it does, I am still waiting for \[\zeta(4)=\frac {\pi^4}{90}\]
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
I see. Let me try and find one of the more elementary proofs.
 2 years ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.