Finding sum of alternating series

[AEJ]

Deduce the sum of the series sum_{n=0}^infinity (-1)^n / (2n+1)^3.

 

[Dr.Peterson]

Keeping in mind that this is not the Free Math Answers forum, but Free Math Help. So, where do you need help?

In particular, what are you supposed to deduce this from? What have you learned that might be of use? What did you try?

 

[pka]

Also, this is the series \(\large{ \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{{{(2n + 1)}^3}}}}} \)
Can you show that the series does converge? If so please post.

 

[AEJ]

Keeping in mind that this is not the Free Math Answers forum, but Free Math Help. So, where do you need help?

In particular, what are you supposed to deduce this from? What have you learned that might be of use? What did you try?

i have tried to replace the value of n which is
=1-1/27+1/125-1/343+1/729-...
=1/1^3 - 1/3^3 + 1/5^3 - 1/7^3 + 1/9^3 -...
=[1-1/3^3] + [1/5^3-1/7^3] + [1/9^3-1/11^3]+...
now how to proceed to find the sum of this infinite series?

 

[AEJ]

Also, this is the series \(\large{ \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{{{(2n + 1)}^3}}}}} \)
Can you show that the series does converge? If so please post.

As n tends to infinity, 1/(2n+1)^3 tends to 0 and also 1/(2n+1)^3 is a decreasing sequence. Therefore it converges.
Now, how to proceed to find the sum of this infinite series?
I have tried to replace the value of n which is
=1-1/27+1/125-1/343+1/729-...
=1/1^3 - 1/3^3 + 1/5^3 - 1/7^3 + 1/9^3 -...
=[1-1/3^3] + [1/5^3-1/7^3] + [1/9^3-1/11^3]+...
Now how to continue?

 

[pka]

To be very clear with you, this is a beast of a summation.
What methods have you studied? Have you done any work with numerical methods.
Here is a computer algebra solution. As you can see the solution involves the \(\zeta\) function.

 

[Dr.Peterson]

We really need to see the context of the problem, which I asked for in part. It said, "deduce", so I deduced that something was said before this that you may be expected to use (though with the result pka showed, that seems less likely than I thought).

Or did the problem you were given only ask about convergence, and you are just assuming you should be able to find the sum?