1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Geometric series and its derivatives

  1. Sep 20, 2015 #1
    1. The problem statement, all variables and given/known data
    I was browsing online and stumbled upon someone's explanation as to why 1 -2 +3 -4 + 5... towards infinity= 1/4. His explanation didn't make sense to me. He starts with a geometric series, takes a derivative, and plugs in for x = -1, and gets a finite value of 1 -2 + 3 - 4 +... = 1/4. It doesn't make sense to me that he would plug in for x = 1, since the series equals 1/(1-x) only for |x|< 1 and converges when |x| <1. So I wanted to know what you guys think.

    2. Relevant equations

    0xn = 1/(1-x), |x| < 1

    3. The attempt at a solution
    This is how the work is shown. He wants to show that 1-2+3-4+5-...=1/4. He begins with:

    0xn = 1/(1-x), |x| < 1
    1 + x + x2 + x3 +... = 1/(1-x)

    Takes a derivative of both sides:

    1 + 2x + 3x2 + 4x3 + ... = 1/(1-x)2

    plugs in for x = -1

    1 -2 +3 -4 + 5... = 1/4

    Are these operations valid?
     
  2. jcsd
  3. Sep 20, 2015 #2

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    No, because the requirement for the infinite sum to converge was |x|<1, which is not obeyed by setting x=-1.
     
  4. Sep 20, 2015 #3
    That is what I thought as well. But apparently, this result is used to prove that 1 + 2 + 3 + 4 +5... = -1/12, and apparently that result is used in physics like String Theory.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted