91Ó°ÊÓ

Express each series as a rational function. $$ \sum_{n=1}^{\infty} \frac{1}{(x-3)^{2 n-1}} $$

Express each series as a rational function. $$ \sum_{n=1}^{\infty}\left(\frac{1}{(x-3)^{2 n-1}}-\frac{1}{(x-2)^{2 n-1}}\right) $$

The exercise explore applications of annuities. Calculate the present values \(P\) of an annuity in which $$\$ 10,000$$ is to be paid out annually for a period of 20 years, assuming interest rates of \(r=0.03, r=0.05,\) and \(r=0.07\).

The exercise explore applications of annuities. Calculate the present values \(P\) of annuities in which $$\$ 9,000$$ is to be paid out annually perpetually, assuming interest rates of \(r=0.03, r=0.05\) and \(r=0.07\).

The exercise explore applications of annuities. Calculate the annual payouts \(C\) to be given for 20 years on annuities having present value $$\$ 100,000$$ assuming respective interest rates of \(r=0.03, r=0.05,\) and \(r=0.07\).

The exercise explore applications of annuities. Calculate the annual payouts \(C\) to be given perpetually on annuities having present value $$\$ 100,000$$ assuming respective interest rates of \(r=0.03, r=0.05,\) and \(r=0.07 .\)

The exercise explore applications of annuities. Suppose that an annuity has a present value \(P=1\) million dollars. What interest rate \(r\) would allow nfor perpetual annual payouts of $$\$ 50,000 ?$$

The exercise explore applications of annuities. Suppose that an annuity has a present value \(P=10\) million dollars. What interest rate \(r\) would allow for perpetual annual payouts of $$\$ 100,000 ?$$

Express the sum of each power series in terms of geometric series, and then express the sum as a rational function. \(x+x^{2}-x^{3}+x^{4}+x^{5}-x^{6}+\cdots \quad\) (Hint: Group powers \(x^{3 k}, x^{3 k-1},\) and \(x^{3 k-2}\).)

Express the sum of each power series in terms of geometric series, and then express the sum as a rational function. \(x+x^{2}-x^{3}-x^{4}+x^{5}+x^{6}-x^{7}-x^{8}+\cdots\) (Hint: Group powers \(x^{4 k}, x^{4 k-1},\) etc.)