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!

Homework Help: Convergent series

  1. Aug 7, 2013 #1
    In my book it says that the series (1+x)^n converges for x<1.

    However I put n = -1 and wolfram says that the series does not converge.

    However if I let x = 1/y where y>1

    then the expansion of (1+1/y)^-1 is equal to: (which I will define as (SERIES 1))
    1 - y + (1/y)2 - (1/y)3 + (1/y)4 - ......

    = 1 + ( (1/y)3 + (1/y)5 + ... ) - ( (1/y)2 + (1/y)4 + ...)

    The series (1/y)3 + (1/y)5 + ... (1/y)3 + 2n + .. is equal to (which I will define as (SERIES 2))
    Ʃ(1/y)3 + 2n where n→∞ and 1≤n<∞.
    I also know that the sum of the (SERIES 2) is less than the series 1 + 1/4 + 1/8 + 1/16 + ... which is convergent therefore (SERIES 2) is convergent.

    Also from (SERIES 1) I know that the sum is positive and therefore the series
    ( (1/y)2 + (1/y)4 + ...) is less than (SERIES 2) + 1 and therefore as the number of terms approaches ∞ the series
    ( (1/y)2 + (1/y)4 + ...) which is positive is less than a [(finite number) + 1] which is a finite number and therefore is convergent.

    ∴Therefore the (SERIES 1) is convergent.


    Am I and the book wrong or is wolfram?
     

    Attached Files:

    • conv.jpg
      conv.jpg
      File size:
      13.9 KB
      Views:
      114
  2. jcsd
  3. Aug 7, 2013 #2

    mfb

    User Avatar
    2017 Award

    Staff: Mentor

    (1+x)^n is not a series.

    If you want to sum that over natural n, it is pointless to set n to anything special (in particular, negative values).

    Do you want to get the taylor expansion at x=0? That converges for x<1 for all n, and for all x for integer n (including 0 if we define 00=1).


    That step requires absolute convergence, so
    1 + |1/y| + |(1/y)2| + |(1/y)3| ......
    has to converge, too.

    That is true for some y only.

    I would expect that the book and WolframAlpha are right. What did you use as query for WA?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted