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    Find the radius of convergence of ∑▒〖(a_n x^n)〗, given a_n=1when n is the square of a natural number and a_n otherwise. If a_n=1 when n=m! for n∈N and a_n=0 otherwise, find the radius of convergence of ∑▒〖(a_n x^n)〗.

    Determine the radius of convergence of ∑▒〖(a_n x^n)〗, if 0<ρ≤|a_n |≤q for all n∈N.

    © BrainMass Inc. brainmass.com December 24, 2021, 8:39 pm ad1c9bdddf
    https://brainmass.com/math/real-analysis/radius-power-series-298876

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    SOLUTION This solution is FREE courtesy of BrainMass!

    To find the radius of the power series , we use the following theorem.
    Theorem: Let , then the radius of convergence is .

    Problem #1
    (a) Find the radius of convergence of , given if is the square of a natural number and otherwise. I think you missed in the posting.
    From the condition, we know that , then we have

    So the radius of convergence is .
    (b) Find the radius of convergence of , given when and otherwise.
    From the condition, we know that , then we have

    So the radius of convergence is .

    Problem #2
    Find the radius of convergence of , if for all .
    From the condition, we know that , then we have

    Here we use the fact that for any positive number , we have . This is because as .
    Thus we must have

    So the radius of convergence is .

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 8:39 pm ad1c9bdddf>
    https://brainmass.com/math/real-analysis/radius-power-series-298876

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