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Chapter 7: Problem 48
Use the Direct Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{4 \sqrt[3]{n}-1} $$
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The ball in Exercise 95 takes the following times for each fall. $$ \begin{array}{ll} s_{1}=-16 t^{2}+16, & s_{1}=0 \text { if } t=1 \\ s_{2}=-16 t^{2}+16(0.81), & s_{2}=0 \text { if } t=0.9 \\ s_{3}=-16 t^{2}+16(0.81)^{2}, & s_{3}=0 \text { if } t=(0.9)^{2} \\ s_{4}=-16 t^{2}+16(0.81)^{3}, & s_{4}=0 \text { if } t=(0.9)^{3} \end{array} $$ \(\vdots\) $$ s_{n}=-16 t^{2}+16(0.81)^{n-1}, \quad s_{n}=0 \text { if } t=(0.9)^{n-1} $$ Beginning with \(s_{2}\), the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is given by \(t=1+2 \sum_{n=1}^{\infty}(0.9)^{n}\) Find this total time.
Consider the formula \(\frac{1}{x-1}=1+x+x^{2}+x^{3}+\cdots\) Given \(x=-1\) and \(x=2\), can you conclude that either of the following statements is true? Explain your reasoning. (a) \(\frac{1}{2}=1-1+1-1+\cdots\) (b) \(-1=1+2+4+8+\cdots\)
Let \(a_{n}=\frac{n+1}{n}\). Discuss the convergence of \(\left\\{a_{n}\right\\}\) and \(\sum_{n=1}^{\infty} a_{n}\).
A fair coin is tossed repeatedly. The probability that the first head occurs on the \(n\) th toss is given by \(P(n)=\left(\frac{1}{2}\right)^{n},\) where \(n \geq 1\) (a) Show that \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=1\). (b) The expected number of tosses required until the first head occurs in the experiment is given by \(\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\) Is this series geometric? (c) Use a computer algebra system to find the sum in part (b).
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. $$ \sum_{n=1}^{\infty} \frac{1}{n(n+1)} $$
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