Chapter 8.4, Problem 6E

In Exercises 3-8, by a closed form we mean an algebraic expression not involving a summation over a range of values or the use of ellipses.

Find a closed form for the generating function for the Sequence $\left\{a_n\right\}$ , where

a) $a_n=-1$ for all $n=0,1,2\dots$

b) $a_n=2^n$ for $n=0,1,2,3,4$ , and $a_0=0$ .

c) $a_n=n-1$ for $n=0,1,2\dots$

d) $a_n=1/\left(n+1\right)!$ for $n=0,1,2\dots$

e) $a_n=\left(\begin{array}{c}n\\ 2\end{array}\right)$ for $n=0,1,2\dots$

f) $a_n=\left(\begin{array}{c}10\\ n+1\end{array}\right)$ for $n=0,1,2\dots$