Finding the Sum of an Infinite Series

[kq6up]

Homework Statement

Find the expectation value of the Energy the Old Fashioned way from example 2.2.

Homework Equations

$\left< E \right> =\frac { 480\hbar ^{ 2 } }{ \pi ^{ 4 }ma^{ 2 } } \sum _{ odds }^{ \infty }{ \frac { 1 }{ { n }^{ 4 } } }$

The Attempt at a Solution

Never mind the details of the physics problem. I am confident of those bits since it is from an example.

Using a symbolic math program, how to I only evaluate odds of a summation? I use sage and Wolfram Alpha normally. I tried using sin(pi*n/2)^2 to eliminate the even terms, but neither program seemed to take well to that.

Thanks,
Chris

 

[STEMucator]

Homework Statement

Find the expectation value of the Energy the Old Fashioned way from example 2.2.

Homework Equations

$\left< E \right> =\frac { 480\hbar ^{ 2 } }{ \pi ^{ 4 }ma^{ 2 } } \sum _{ odds }^{ \infty }{ \frac { 1 }{ { n }^{ 4 } } }$

The Attempt at a Solution

Never mind the details of the physics problem. I am confident of those bits since it is from an example.

Using a symbolic math program, how to I only evaluate odds of a summation? I use sage and Wolfram Alpha normally. I tried using sin(pi*n/2)^2 to eliminate the even terms, but neither program seemed to take well to that.

Thanks,
Chris

If I'm reading correctly you only want the sum of the odd terms. So make a change of index. Suppose that $n = 2k+1$. Then:


$\left< E \right> =\frac { 480\hbar ^{ 2 } }{ \pi ^{ 4 }ma^{ 2 } } \sum _{ k=0 }^{ \infty }{ \frac { 1 }{ { (2k+1) }^{ 4 } } }$

 

[kq6up]

That might work. Let me give that a shot.

Chris

 

[kq6up]

Perfect, thank you.

Chris