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Chapter 8: Problem 6
What is the condition for convergence of the geometric series \(\sum_{k=0}^{\infty} a r^{k} ?\)
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Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. $$a_{n+1}=\frac{1}{2} a_{n}+2 ; a_{0}=5$$
Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 3^{-k}$$
The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined by \(\zeta(x)=\sum_{k=1}^{\infty} \frac{1}{k^{x}} .\) When \(x\) is a real number, the zeta function becomes a \(p\) -series. For even positive integers \(p,\) the value of \(\zeta(p)\) is known exactly. For example, $$\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}, \quad \sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90}, \quad \text { and } \quad \sum_{k=1}^{\infty} \frac{1}{k^{6}}=\frac{\pi^{6}}{945}, \ldots$$ Use the estimation techniques described in the text to approximate \(\zeta(3)\) and \(\zeta(5)\) (whose values are not known exactly) with a remainder less than \(10^{-3}\).
A heifer weighing 200 lb today gains 5 lb per day with a food cost of \(45 \mathrm{c} /\) day. The price for heifers is \(65 \mathrm{q} / \mathrm{lb}\) today but is falling \(1 \% /\) day. a. Let \(h_{n}\) be the profit in selling the heifer on the \(n\) th day, where \(h_{0}=(200 \mathrm{lb}) \cdot(\$ 0.65 / \mathrm{lb})=\$ 130 .\) Write out the first 10 terms of the sequence \(\left\\{h_{n}\right\\}\). b. How many days after today should the heifer be sold to maximize the profit?
Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{75^{n-1}}{99^{n}}+\frac{5^{n} \sin n}{8^{n}}$$
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