### Author Topic: Q4: TUT 0201  (Read 1777 times)

#### Ziyi Wang

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##### Q4: TUT 0201
« on: February 17, 2020, 07:12:50 PM »
Question: Find the radius of convergence of the given power series.
$$\sum^\infty _{k=0} \frac{(k!)^2}{(2k)!}(z-2)^k$$
Answer: By ratio test, we have that:
$$\frac{1}{R} = lim_{k\rightarrow \infty} |\frac{\frac{((k+1)!)^2}{(2(k+1))!}}{\frac{(k!)^2}{(2k)!}}|$$
$$= lim_{k\rightarrow \infty} |\frac{\frac{(k+1)^2(k)^2(k-1)^2...}{(2k+2)(2k+1)(2k)...}}{\frac{(k)^2(k-1)^2...}{(2k)(2k-1)...}}|$$
$$= lim_{k\rightarrow \infty} |\frac{(k+1)^2}{(2k+2)(2k+1)}|$$
$$= lim_{k\rightarrow \infty} \frac{k+1}{4k+2}$$
By L’Hospital’s Rule, we know that
$$\frac{1}{R} = lim_{k\rightarrow \infty} \frac{k+1}{4k+2} = lim_{k\rightarrow \infty} \frac{1}{4} = \frac{1}{4}$$
Therefore, we conclude that $R=4$.