Problem 96
The terms of a series \(\sum_{n=1}^{\infty} a_{n}\) are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning. $$ a_{1}=\frac{1}{4}, a_{n+1}=\sqrt[n]{a_{n}} $$
Problem 97
In Exercises 97-100, use the Ratio Test or the Root Test to determine the convergence or divergence of the series. $$ 1+\frac{1 \cdot 2}{1 \cdot 3}+\frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5}+\frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7}+\cdots $$
Problem 97
Consider the sequence \(\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots\) (a) Compute the first five terms of this sequence. (b) Write a recursion formula for \(a_{n}, n \geq 2\). (c) Find \(\lim _{n \rightarrow \infty} a_{n}\).
Problem 97
Probability In Exercises 97 and 98, the random variable \(\boldsymbol{n}\) represents the number of units of a product sold per day in a store. The probability distribution of \(n\) is given by \(P(n) .\) Find the probability that two units are sold in a given day \([P(2)]\) and show that \(P(1)+P(2)+P(3)+\cdots=1\). $$ P(n)=\frac{1}{2}\left(\frac{1}{2}\right)^{n} $$
Problem 98
The random variable \(\boldsymbol{n}\) represents the number of units of a product sold per day in a store. The probability distribution of \(n\) is given by \(P(n) .\) Find the probability that two units are sold in a given day \([P(2)]\) and show that \(P(1)+P(2)+P(3)+\cdots=1\). $$ P(n)=\frac{1}{3}\left(\frac{2}{3}\right)^{n} $$
Problem 98
Compute the first six terms of the sequence \(\left\\{a_{n}\right\\}=\left\\{\left(1+\frac{1}{n}\right)^{n}\right\\}\) If the sequence converges, find its limit.
Problem 98
Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. $$ 1+\frac{2}{3}+\frac{3}{3^{2}}+\frac{4}{3^{3}}+\frac{5}{3^{4}}+\frac{6}{3^{5}}+\cdots $$
Problem 99
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).
Problem 99
Suppose that \(\sum a_{n}\) and \(\sum b_{n}\) are series with positive terms. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) and \(\sum b_{n}\) converges, \(\Sigma a_{n}\) also converges.
Problem 99
Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. $$ \frac{1}{(\ln 3)^{3}}+\frac{1}{(\ln 4)^{4}}+\frac{1}{(\ln 5)^{5}}+\frac{1}{(\ln 6)^{6}}+\cdots $$
