b7-power series: problem 5\n(1 point)\nfind the interval of convergence for the power series\n sum_{n =…

Explanation:

Step1: Apply Ratio Test

For a power series (\sum_{n = 0}^{\infty}a_{n}(x - c)^{n}), here (a_{n}=n!) and (c = 10). Use the ratio test: (L=\lim_{n\rightarrow\infty}\left|\frac{a_{n + 1}(x - c)^{n+1}}{a_{n}(x - c)^{n}}\right|). Substitute (a_{n}=n!) and (a_{n+1}=(n + 1)!): [ \begin{align*} L&=\lim_{n\rightarrow\infty}\left|\frac{(n+1)!(x - 10)^{n+1}}{n!(x - 10)^{n}}\right|\ &=\lim_{n\rightarrow\infty}\left|(n + 1)(x-10)\right| \end{align*} ]

Step2: Analyze the limit

If (x\neq10), then (\lim_{n\rightarrow\infty}\left|(n + 1)(x - 10)\right|=\infty) (since (\lim_{n\rightarrow\infty}(n+1)=\infty) for non - zero ((x - 10))). The series converges when (L<1). But for (x\neq10), (L=\infty> 1). When (x = 10), the series becomes (\sum_{n=0}^{\infty}n!(10 - 10)^{n}=\sum_{n = 0}^{\infty}n!\times0^{n}=0+0+\cdots) (since (0^{n}=0) for (n\geq1) and (0!0^{0}=1\times1 = 1), but the sum is just (1+0+0+\cdots)).

Answer:

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