Question

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

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

Answer 1

series converges for all values of x.

Therefore radius of convergence of series is R=+∞

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)!rk k!(k+q)! (k+p+1). => lim •'(k+1)(k+9+1)<1 k(1+&+ 1). => klima 'k(1+1)*(1+7+ D' (1 + + ). => klima (1 + Đ(i+1+' 0(1+0+ 0.1 (1+0)(1+0+0) => 0 <1

=>series converges for all values of x

Therefore radius of convergence of series is R=+∞

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