Find the radius of convergence of:
$$\sum \frac{(K/e)^{3K}}{3K!}(7x)^K.$$
I figured that the ratio test could be used because of the factorial. Can can someone help me please?
2026-04-04 12:07:30.1775304450
Finding the radius of convergence
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$$a_k:=\frac{k^{3k}}{e^{3k}(3k)!}(7x)^k\Longrightarrow\left|\frac{a_{k+1}}{a_k}\right|=7|x|\frac{(k+1)^{3k+3}}{e^{3k+3}(3k+3)!}\frac{e^{3k}(3k)!}{k^{3k}}=$$
$$=\frac{(k+1)^3}{(3k+1)(3k+2)(3k)}\left(1+\frac{1}{k}\right)^{3k}\frac{7}{e^3}|x|\xrightarrow[k\to\infty]{}\frac{1}{27}\cdot e^3\cdot\frac{7}{e^3}|x|=\ldots$$
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