1. Not finding help here? Sign up for a free 30min 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!

Alternating Series Estimation Theorem

  1. May 6, 2017 #1
    1. The problem statement, all variables and given/known data
    Using the power series for ln(x + 1) and the Estimation Theorem for the Alternating Series, we conclude that the least number of terms in the series needed to approximate ln 2 with error < 3/1000 is: (i) 333 (ii) 534 (iii) 100 (iv) 9 (v) 201

    2. Relevant equations
    ln(x+1) = Σ(-1)^nx^n/n!

    3. The attempt at a solution
    I know that the alternating series estimation thm is |Sn-S| ≤ (estimation) which is 3/1000. I get x to be equal to 1 since we want ln(2), but when I setup the equation I get lost on how to simplify this to a specific n value. (Calculators are not allowed on exam so I am rusty with algebra).

    I get (-1)^n+1 * 2^n/(n+1)! ≤ 3/1000 which gives 2^n ≤ 3/1000 * (n+1) and I can't figure how to get the n in the exponent down without using ln yet the answers are specific numbers.
     
  2. jcsd
  3. May 6, 2017 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    For an alternating series, the estimated error is less than the magnitude of the term following the last one.
    What is the Taylor series of ln(x+1) around x=0?
     
  4. May 6, 2017 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    (1) You have the wrong series; your series is the expansion of ##e^{-x}##, which is not what was asked. The expansion of ##\ln(1+x)## is a lot simpler, and produces a problem solvable easily in an exam setting without a calculator.
    (2) You say you need ##x = 1##, so how did ##1^n## become ##2^n##?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted