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

In summary, the conversation discusses finding the interval of convergence for the function f'(x), which is the derivative of f(x). The function is a series with n starting at 1 and going to infinity, and involves the terms (x-5)^n and (-1)^n. There is some confusion about taking the derivative with n's and x's, but it is eventually resolved that n can be treated as a constant. It is also mentioned that when taking the derivative of a series, one must be careful to avoid getting a negative power for x. Finally, it is determined that finding the interval of convergence for the original function should be enough, as taking the derivative does not change the interval of convergence.
  • #1
bcjochim07
374
0

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.
 
Physics news on Phys.org
  • #2
Is the derivative the sum from n=1 to infinity of [(x-5)^(n-1) * (-1)^(n+1)]/[5^n]?
 
Last edited:
  • #3
bcjochim07 said:
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)?
 
  • #4
Oh... it should be (-1)^(n+1) in the original function, I just typed it out wrong.
 
  • #5
It looks fine then.
 
  • #6
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.
 
  • #7
I think finding R for the original function should be enough; differentiation does not change R.

Using this way, you get R = 5?
 

1. What is a power series?

A power series is a type of mathematical series that is written in the form of a polynomial, where the powers of x increase by one in each term. It is typically used to represent functions as infinite sums of powers of x.

2. What is the interval of convergence for a power series?

The interval of convergence for a power series is the range of x-values for which the series will converge, or have a finite sum. This interval can be determined by using various convergence tests, such as the ratio or root test.

3. How do you find the interval of convergence for f'(x) in a power series?

To find the interval of convergence for f'(x) in a power series, you can first find the interval of convergence for the original power series, and then take the derivative of the series to find the interval for f'(x). This can be done by using the same convergence tests as before, but with the derivative of the original function.

4. What are some common convergence tests used for finding the interval of convergence?

Some common convergence tests used for finding the interval of convergence include the ratio test, the root test, and the alternating series test. These tests can help determine whether a series will converge or diverge, and can also be used to find the interval of convergence.

5. Can the interval of convergence for a power series change?

Yes, the interval of convergence for a power series can change depending on the function being represented. It is important to always check for convergence and determine the interval for a specific function, rather than assuming it will be the same as a previous series with similar terms.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
88
  • Calculus and Beyond Homework Help
Replies
2
Views
658
  • Calculus and Beyond Homework Help
Replies
1
Views
89
  • Calculus and Beyond Homework Help
Replies
26
Views
821
  • Calculus and Beyond Homework Help
Replies
7
Views
653
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
164
  • Calculus and Beyond Homework Help
Replies
22
Views
3K
  • Calculus and Beyond Homework Help
Replies
3
Views
355
  • Calculus and Beyond Homework Help
Replies
2
Views
61
Back
Top