Chapter 7: Problem 2
Write the first five terms of the sequence. \(a_{n}=\frac{3^{n}}{n !}\)
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
Find the sum of the convergent series. $$ 1+0.1+0.01+0.001+\cdots $$
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 \(\rho\).
Find the sum of the convergent series. $$ \sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}-\frac{1}{3^{n}}\right) $$
Writing In Exercises 89 and 90 , use a graphing utility to determine the first term that is less than 0.0001 in each of the convergent series. Note that the answers are very different. Explain how this will affect the rate at which each series converges. $$ \sum_{n=1}^{\infty} \frac{1}{n(n+1)} $$
Find the values of \(x\) for which the infinite series \(1+2 x+x^{2}+2 x^{3}+x^{4}+2 x^{5}+x^{6}+\cdots\) converges. What is the sum when the series converges?
What do you think about this solution?
We value your feedback to improve our textbook solutions.
