Question

7. Show that if p and q are positive integers, then the radius of convergence of the power series below is R= too +00 k=2

Show that if p and q are positive integers, then the radius of convergence of the power series below is R= +∞

\(\sum_{\mathrm{k}=2}^{+\infty} \frac{(\mathrm{k}+\mathrm{p}) !}{\mathrm{k} !(\mathrm{k}+\mathrm{q}) !} \mathrm{x}^{\mathrm{k}}\)

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Answer #1

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series converges for all values of x.

Therefore radius of convergence of series is R=+∞


answered by: Zahidul Hossain
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Answer #2

Ratio test:

(k + p)!* k!(k+q)!** (k+p+1) +1 (k + 1)!(k+ 9 + 1)!
(k+p+1)!r*+1 (k+1)!(k+q+1)! => lim k-> (k+p)!***|<1 k!(k+q)! (k+p+1)/(k+p)!).r4+1 (k+1)(k!)(k+q+1)(k+q)! => lim k-> o (k+p)!
=>series converges for all values of x

Therefore radius of convergence of series is R=+∞

please comment if you have any doubt.please rate if helpful

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