In the event that a series converges uniformly, one can consider the derivative of the series to arrive at the summation of other infinite series.

a. Differentiate the series representation for \(f(x)=\frac{1}{1-x}\) to sum the series \(\sum_{n=1}^{\infty} n x^{n},|x|<1\).

b. Use the result from part a to sum the series \(\sum_{n=1}^{\infty} \frac{n}{5^{n}}\).

c. Sum the series \(\sum_{n=2}^{\infty} n(n-1) x^{n},|x|<1\).

d. Use the result from part c to sum the series \(\sum_{n=2}^{\infty} \frac{n^{2}-n}{5^{n}}\).

e. Use the results from this problem to sum the series \(\sum_{n=4}^{\infty} \frac{n^{2}}{5^{n}}\).