Finding the Interval of Convergence for f'(x) in a Power Series - Homework Help

[bcjochim07]

Homework Statement

Find the interval of convergence of f'(x)

f(x) Sum from n=1 to infinity [(x-5)^n*(-1)^n]/[n5^n]

Homework Equations

The Attempt at a Solution

My problem is I am unsure how to take the derivative with the n's and x's should I treat n as a constant?

After that, I think I can get the interval of convergence.

 

[bcjochim07]

Is the derivative the sum from n=1 to infinity of [(x-5)^(n-1) * (-1)^(n+1)]/[5^n]?

 

[Dick]

Is the derivative the sum from n=1 to infinity of [(x-5)^(n-1) * (-1)^(n+1)]/[5^n]?

Almost. How did (-1)^n become (-1)^(n+1)?

 

[bcjochim07]

Oh... it should be (-1)^(n+1) in the original function, I just typed it out wrong.

 

[Dick]

It looks fine then.

 

[rostbrot]

yes you just treat n as a constant.

Be careful when taking the derivative of a series though. Here the issue didn't come up, but if you have x^n where n starts at 0 and end up with x^(n-1) where n starts at 0 then you would get x^(-1) for n=0 which is a no-no so you'd have to move you n up to starting at 1 do solve that problem.

 

[rootX]

I think finding R for the original function should be enough; differentiation does not change R.

Using this way, you get R = 5?