Question

3. Find the limits of the following sequences, or show that the limit does not exist. \text { (a) }\left\{a_{k}\right\}_{k=1}^{\infty} \text { where } a_{k}=\frac{b_{1}}{b_{k-1}} \text { and } b_{k}=\frac{(2

k) !}{h !^{2}} \text { (c) }\left\{a_{n}\right\}_{n=1}^{\infty} a_{n}=\frac{1}{2}\left(a_{n-1}-\frac{4}{a_{n-1}}\right) a_{n}=\frac{n^{3}-n^{2}+n-1}{\sqrt{n^{2}+n+1}-\sqrt{n}} \text { (d) }\left\{a_{n}\right\}_{n=1}^{\infty} \text { where } where a1 = 1 and an is defined by the recurrence \text { (b) }\left\{a_{k}\right\}_{k=1}^{\infty} where a1 = 2 and an is defined by the recurrence a_{n}=\frac{1}{3}\left(2 a_{n-1}+\frac{2}{a_{n-1}^{2}}\right)

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