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Thread: radius of convergence

  1. #1
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    radius of convergence

    I was trying to get the radius of convergence for the following question but get stuck along the way.

    $\displaystyle \sum_{n=0}^\infty \frac{(n!)^k}{(kn)!}x^n$

    By using ratio test:

    $\displaystyle lim_{n\rightarrow\infty}|{\frac{(n+1)!^k}{(k(n+1)) !}x^{n+1}/\frac{(n!)^k}{(kn)!}x^n}| <1$

    but i can't seem to simplify $\displaystyle \frac{(kn)!}{k(n+1)!}$

    The answer is: $\displaystyle k^k$
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  2. #2
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    Quote Originally Posted by noob mathematician View Post
    I was trying to get the radius of convergence for the following question but get stuck along the way.

    $\displaystyle \sum_{n=0}^\infty \frac{(n!)^k}{(kn)!}x^n$

    By using ratio test:

    $\displaystyle lim_{n\rightarrow\infty}|{\frac{(n+1)!^k}{(k(n+1)) !}x^{n+1}/\frac{(n!)^k}{(kn)!}x^n}| <1$

    but i can't seem to simplify $\displaystyle \frac{(kn)!}{k(n+1)!}$

    The answer is: $\displaystyle k^k$

    Sure that is the answer. You began correctly but stopped short of opening and simplifying stuff a little bit more:

    $\displaystyle \frac{(kn)!}{(k(n+1))!}=\frac{(kn)!}{(kn+k)!}=\fra c{1}{(kn+1)(kn+2)\cdot...\cdot(kn+k)}$ $\displaystyle =\frac{1}{k^kn^k+\{\mbox{terms with n to a power less than k }\}}$

    Now go back to your limit and plug in the above and you'll get the correct answer.

    Tonio
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  3. #3
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    Quote Originally Posted by noob mathematician View Post
    I was trying to get the radius of convergence for the following question but get stuck along the way.

    $\displaystyle \sum_{n=0}^\infty \frac{(n!)^k}{(kn)!}x^n$

    By using ratio test:

    $\displaystyle lim_{n\rightarrow\infty}|{\frac{(n+1)!^k}{(k(n+1)) !}x^{n+1}/\frac{(n!)^k}{(kn)!}x^n}| <1$

    but i can't seem to simplify $\displaystyle \frac{(kn)!}{k(n+1)!}$

    The answer is: $\displaystyle k^k$
    $\displaystyle a_n = \frac{{{{\left( {n!} \right)}^k}}}{{\left( {kn} \right)!}}{x^n} \Rightarrow {a_{n + 1}} = \frac{{{{\left( {\left( {n + 1} \right)!} \right)}^k}}}{{\left( {kn + k} \right)!}}{x^{n + 1}} = \frac{{{{\left( {n!} \right)}^k}{{\left( {n + 1} \right)}^k}}}{{\left( {kn} \right)!\left( {kn + 1} \right)\left( {kn + 2} \right) \ldots \left( {kn + k} \right)}}{x^{n + 1}}.$

    $\displaystyle \frac{{{a_{n + 1}}}}{{{a_n}}} = \frac{{x{{\left( {n + 1} \right)}^k}}}{{\left( {kn + 1} \right)\left( {kn + 2} \right) \ldots \left( {kn + k} \right)}} = \frac{{x{{\left( {n + 1} \right)}^k}}}{{\prod\limits_{i = 1}^k {\left( {kn + i} \right)} }}.$

    $\displaystyle \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{a_{n + 1}}}}
    {{{a_n}}}} \right| = \left| x \right|\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {n + 1} \right)}^k}}}{{\prod\limits_{i = 1}^k {\left( {kn + i} \right)} }} = \left| x \right|\mathop {\lim }\limits_{n \to \infty } \frac{{{n^k}{{\left( {1 + \frac{1}{n}} \right)}^k}}}{{{k^k}{n^k}\prod\limits_{i = 1}^k {\left( {1 + \frac{i}{{kn}}} \right)} }} =$

    $\displaystyle = \frac{{\left| x \right|}}
    {{{k^k}}}\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {1 + \frac{1}
    {n}} \right)}^k}}}
    {{\prod\limits_{i = 1}^k {\left( {1 + \frac{i}
    {{kn}}} \right)} }} = \frac{{\left| x \right|}}
    {{{k^k}}} \cdot \frac{{{{\left( {1 + 0} \right)}^k}}}
    {{\prod\limits_{i = 1}^k {\left( {1 + 0} \right)} }} = \frac{{\left| x \right|}}
    {{{k^k}}}.$

    $\displaystyle \frac{{\left| x \right|}}{{{k^k}}} < 1 \Leftrightarrow - {k^k} < x < {k^k} \Rightarrow R = k^k.$
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