Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Power series

  1. Mar 28, 2017 #1
    I've 2 questions
    1) Why do we take absolute of the power series?
    2) I don't get why the interval of convergence is from -inifinity to +infinity. You can find the problem below.

    upload_2017-3-28_17-44-24.png
     

    Attached Files:

  2. jcsd
  3. Mar 28, 2017 #2

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    For your first question, look up the "ratio test".

    If the series converges for all ##x## then by definition the radius of convergence is ##\infty##.
     
  4. Mar 28, 2017 #3

    FactChecker

    User Avatar
    Science Advisor
    Gold Member

    That is called the ratio test of convergence. (https://en.wikipedia.org/wiki/Ratio_test) The absolute value is all you need to test convergence.
    It converges if the absolute value of the limit is less than 1. In this case, the limit of the ratio is 0 regardless of the value of x. So it converges for any value of x.
     
  5. Mar 28, 2017 #4
    I've seen different problems with the ratio test and they didn't use absolute but when it comes to power series, they use it. Why?
     
  6. Mar 28, 2017 #5

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    If you don't take the absolute value and the series has either alternating terms or at least infinitely many positive and negative terms, then the limit won't exist - unless it's 0. The test is better with the absolute value.
     
  7. Mar 28, 2017 #6

    Mark44

    Staff: Mentor

    A power series is a series in powers of a variable such as x. The Ratio Test requires that all terms be positive, but with a variable, some terms could be negative, if x is negative. The problems you've seen that didn't use absolute values were almost certainly series in which all the terms were positive, such as ##\sum_{n = 1}^\infty \frac 1 {n^2 + 1}##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Power series
  1. Power series (Replies: 3)

  2. Power series (Replies: 2)

  3. Power series (Replies: 2)

  4. Power series (Replies: 4)

  5. Power Series (Replies: 4)

Loading...