91Ó°ÊÓ

Write the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{3^{n}}{2^{n-1}}=3-\frac{9}{2}+\frac{27}{4}-\frac{81}{8}+\frac{243}{16}-\cdots $$

Write the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n !}=1-\frac{1}{2}+\frac{1}{6}-\frac{1}{24}+\frac{1}{120}-\cdots $$

Verify that the geometric series converges. $$ \sum_{n=0}^{\infty}(0.9)^{n}=1+0.9+0.81+0.729+\cdots $$

Determine the convergence or divergence of the sequence. If the sequence converges, find its limit. $$ a_{n}=\frac{n !}{(n+1) !} $$

Use Newton’s Method to obtain a general rule for approximating the indicated radical. \(\sqrt{a}\left[\text {Hint}: \text { Consider } f(x)=x^{2}-a .\right]\)

Annuity A deposit of 100 dollars is made at the beginning of each month for 5 years in an account that pays \(10 \%\) interest, compounded monthly. Use a symbolic algebra utility to find the balance \(A\) in the account at the end of the 5 years. $$A=100\left(1+\frac{0.10}{12}\right)+\cdots+100\left(1+\frac{0.10}{12}\right)^{60}$$

Annuity A deposit of \(P\) dollars is made every month for \(t\) years in an account that pays an annual interest rate of \(r \%,\) compounded monthly. Let \(N=12 t\) be the total number of deposits. Show that the balance in the account after \(t\) years is $$A=P\left[\left(1+\frac{r}{12}\right)^{N}-1\right]\left(1+\frac{12}{r}\right), \quad t>0$$

Depreciation A company buys a machine for 225,000 dollars that depreciates at a rate of \(30 \%\) per year. Find a formula for the value of the machine after \(n\) years. What is its value after 5 years?

Inflation Rate If the average price of a new car increases \(2.5 \%\) per year and the average price is currently 28,400 dollars, then the average price after \(n\) years is \(P_{n}= 28,400(1.025)^{n} .\) Compute the average prices for the first 5 years of increases.

Cost A well-drilling company charges 25 dollars for drilling the first foot of a well, 25.10 dollars for drilling the second foot, 25.20 dollars for the third foot, and so on. Determine the cost of drilling a 100 -foot well.