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Chapter 7: Problem 44
Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} n\left(\frac{3}{2}\right)^{n} $$
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A company buys a machine for \(\$ 225,000\) 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?
Use the formula for the \(n\) th partial sum of a geometric series $$\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}$$ You go to work at a company that pays \(\$ 0.01\) for the first day, \(\$ 0.02\) for the second day, \(\$ 0.04\) for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working (a) 29 days, (b) 30 days, and (c) 31 days?
Suppose that \(\sum a_{n}\) and \(\sum b_{n}\) are series with positive terms. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\infty\) and \(\sum b_{n}\) diverges, \(\sum a_{n}\) also diverges.
Consider the sequence \(\left\\{a_{n}\right\\}\) where \(a_{1}=\sqrt{k}, a_{n+1}=\sqrt{k+a_{n}}\), and \(k>0\) (a) Show that \(\left\\{a_{n}\right\\}\) is increasing and bounded. (b) Prove that \(\lim _{n \rightarrow \infty} a_{n}\) exists. (c) Find \(\lim _{n \rightarrow \infty} a_{n}\).
In your own words, define each of the following. (a) Sequence (b) Convergence of a sequence (c) Monotonic sequence (d) Bounded sequence
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