Do the following: a. Compute: (lim _{n ightarrow infty} n ln left(1-frac{3}{n} ight)). b. Use L'Hopital's Rule to evaluate (L=lim _{x ightarrow infty}left(1-frac{4}{x}

Do the following:

a. Compute: \(\lim _{n \rightarrow \infty} n \ln \left(1-\frac{3}{n}\right)\).

b. Use L'Hopital's Rule to evaluate \(L=\lim _{x \rightarrow \infty}\left(1-\frac{4}{x}\right)^{x}\). [Consider \(\ln L\).

c. Determine the convergence of \(\sum_{n=1}^{\infty}\left(\frac{n}{3 n+2}\right)^{n^{2}}\).

d. Sum the series \(\sum_{n=1}^{\infty}\left[\tan ^{-1} n-\tan ^{-1}(n+1)\right]\) by first writing the \(N\) th partial sum and then computing \(\lim _{N \rightarrow \infty} s_{N}\).