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Math Help - series convergence and radius of convergence

  1. #1
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    series convergence and radius of convergence

    Can you help mr to solve that
    let c0,c1,c2...in R prove that if lim when n goes to infinity of abs(c(subn)/c(sub n+1)) exists,it is equal to the radius of convergence of the power series c(subn)x^n
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by drpawel View Post
    Can you help mr to solve that
    let c0,c1,c2...in R prove that if lim when n goes to infinity of abs(c(subn)/c(sub n+1)) exists,it is equal to the radius of convergence of the power series c(subn)x^n
    Ok let \left\{c_n\right\} be a sequence in \mathbb{R}. Ok let us split this up into three cases: \lim_{n\to\infty}\frac{c_{n}}{c_{n+1}}=L\ne{0}, \lim_{n\to\infty}\frac{c_{n}}{c_{n+1}}=\infty, and \lim_{n\to\infty}\frac{c_{n}}{c_{n+1}}=0.

    Case 1

    Suppose we have a series of the form \sum{c_nx^n}. Then to determine the radius of convergence I apply the Ratio test. So \lim_{n\to\infty}\left|\frac{c_{n+1}x^{n+1}}{c_{n}  x^n}\right|=|x|\lim_{n\to\infty}\frac{c_{n+1}}{c_n  }<1. So now by our assumption above this is equivalent to |x|\frac{1}{L}<1\implies|x|<L, thus the radius of convergence of case 1 is L and we are done.

    Case 2
    Now suppose we have the same series but we know have that \lim_{n\to\infty}|x|\frac{c_{n}}{c_{n+1}}\implies|  x|\cdot{0}<1, which is an identity for |x|<\infty. Thus the radius of convergence is \infty.

    Case 3
    Now once again we reach that |x|\lim_{n\to\infty}\frac{c_n}{c_{n+1}}=|x|\cdot{\  infty}<1 which only occurs when x=0.

    The last two I did more holistically than the first, if you need more rigor I leave that up to you.
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