1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

A logarithmic convergence tests - Analysis

  1. May 8, 2014 #1
    1. The problem statement, all variables and given/known data

    Given a non-negative sequence [itex]\{a_{n}\}_{n=1}^{\infty}[/itex]. Proove that the serie [itex]\Sigma_{n=1}^{\infty}a_{n}[/itex] converge if and only if [itex]\Sigma_{n=1}^{\infty}\ln(1+a_{n})[/itex] converges.

    2. Relevant equations

    3. The attempt at a solution

    My first attempt is the direct comparison test. Which tells that if [itex]\Sigma_{n=1}^{\infty}\ln(1+a_{n})[/itex] converges and if there exist a reel number γ so [itex]a_{n}\leq\gamma\ln(1+a_{n})[/itex] for all n, then the series [itex]\Sigma_{n=1}^{\infty}a_{n}[/itex] also converges.

    But i can't find such a γ.
    Last edited: May 8, 2014
  2. jcsd
  3. May 8, 2014 #2
    Hint*: Note that ##\sum_{n=1}^\infty ln(1+a_n)## is a sum of logarithms. What can you do to change logarithms into multiplication?

  4. May 8, 2014 #3
    yeah, so [itex]\Sigma_{n=1}^{\infty}\ln(1+a_{n}) = \ln\Big[\Pi_{n=1}^{\infty}(1+a_{n})\Big][/itex]. But how am I supposed to conclude from this, that the series [itex]\Sigma_{n=1}^{\infty}a_{n}[/itex] must then also converge ?
  5. May 8, 2014 #4
    Well we know that ##\sum_{n=1}^\infty ln(1+a_n)=C##, where C is a constant, since the sum converges. And we know ##\sum_{n=1}^\infty ln(1+a_n)=ln[\prod_{n=1}^\infty(1+a_n)]=C##. So if we take the exponent of both sides we get ##\prod_{n=1}^\infty(1+a_n)=e^C##. If you expand the product what do you notice? Now you have a bound for the sum ##\sum_{n=1}^\infty a_n ##
  6. May 8, 2014 #5


    User Avatar
    Homework Helper

    Why is that if every [itex]a_n \geq 0[/itex], we have that [tex]\exp\left(\sum_{n=1}^N \ln(1 + a_n)\right) = \prod_{n=1}^N (1 + a_n) \geq 1 + \sum_{n=1}^N a_n \geq 1
    [/tex] for every [itex]N \in \mathbb{N}[/itex]? (Hint: consider what happens if you expand the product.)

    If [itex]\sum_{n=1}^\infty \ln(1 + a_n)[/itex] converges, what can you say about the convergence of [itex]\sum_{n=1}^\infty a_n[/itex]?

    Conversely, if [itex]\sum_{n=1}^\infty a_n[/itex] diverges, what can you conclude about the convergence of [itex]\sum_{n=1}^\infty \ln(1 + a_n)[/itex]?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - logarithmic convergence tests Date
Improper integral convergence from 0 to 1 Yesterday at 4:59 PM
Logarithmic integration Jan 27, 2018
Discrete logarithm property Jan 16, 2018
Find the derivative of this function Jan 12, 2018
Using logarithms in vector Calculus Jan 8, 2018