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Chapter 7: Problem 6
Find the first five terms of the sequence of partial sums. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n !} $$
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Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. \(1170-\) ca. 1240 ) encountered the sequence now bearing his name. It is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}, \quad\) where \(\quad a_{1}=1\) and \(a_{2}=1\) (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by \(b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1\) (c) Using the definition in part (b), show that $$ b_{n}=1+\frac{1}{b_{n-1}} $$ (d) The golden ratio \(\rho\) can be defined by \(\lim _{n \rightarrow \infty} b_{n}=\rho .\) Show that \(\rho=1+1 / \rho\) and solve this equation for \(\rho\).
A ball is dropped from a height of 16 feet. Each time it drops \(h\) feet, it rebounds \(0.81 h\) feet. Find the total distance traveled by the ball.
Compute the first six terms of the sequence \(\left\\{a_{n}\right\\}=\\{\sqrt[n]{n}\\} .\) If the sequence converges, find its limit.
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 \(\lim _{n \rightarrow \infty} a_{n}\).
Find all values of \(x\) for which the series converges. For these values of \(x,\) write the sum of the series as a function of \(x\). $$ \sum_{n=0}^{\infty} 4\left(\frac{x-3}{4}\right)^{n} $$
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