(Solved): 1. ( 8 points) Show that the following sequences are divergent. (a) \( \left(\sin \frac{n \pi}{4}\r ...

1. ( 8 points) Show that the following sequences are divergent. (a) \( \left(\sin \frac{n \pi}{4}\right) \) (b) \( \left(\left(1+(-1)^{n}\right) \sqrt{n}\right) \) 2. (12 points) We know \( \lim \left(x_{n}\right)=e \) where \( x_{n}=\left(1+\frac{1}{n}\right)^{n} \). Consider the subsequences of \( \left(x_{n}\right) \). For example, \( \left((1+1 /(2 k))^{2 k}\right) \) is just the subsequence \( \left(x_{2 k}\right) \), and hence it is convergent to \( e \). Find the following limits. (a) \( \lim \left(\left(1+\frac{1}{2 k-1}\right)^{2 k}\right) \) (b) \( \lim \left(\left(1+\frac{1}{k^{2}}\right)^{2 k^{2}}\right) \) (c) \( \lim \left(\left(1-\frac{1}{2 k+1}\right)^{2 k}\right) \).