Chapter 8.4, Problem 7E

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.

For each of these generating functions, provide a dosed formula for the sequence it determines.

a) ${\left(3x+4\right)}^3$

b) ${\left(x^2+1\right)}^3$

c) $1/\left(1-5x\right)$

d) $x^3/\left(1+3x\right)$

e) $x^2+3z+7+\left(1/\left(1-x^2\right)\right)$

f) $\left(x^4/\left(1-x^4\right)\right)-x^3-x^2-x-1$

g) $x^2/{\left(1-x\right)}^2$

h) $2e^{2x}$