91Ó°ÊÓ

In Exercises \(45-48,\) verify the sum. Then use a graphing utility to approximate the sum with an error of less than 0.0001 $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n}=\ln 2 $$

(a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0.2 \overline{15} $$

Prove that if \(f\) is an even function, then its \(n\) th Maclaurin polynomial contains only terms with even powers of \(x\)

Bessel Function The Bessel function of order 0 is \(J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{2 k}}{2^{2 k}(k !)^{2}}\) (a) Show that the series converges for all \(x\). (b) Show that the series is a solution of the differential equation \(x^{2} J_{0}^{\prime \prime}+x J_{0}^{\prime}+x^{2} J_{0}=0 .\) (c) Use a graphing utility to graph the polynomial composed of the first four terms of \(J_{0}\) - (d) Approximate \(\int_{0}^{1} J_{0} d x\) accurate to two decimal places.

Describe the difference between \(\lim _{n \rightarrow \infty} a_{n}=5\) and \(\sum_{n=1}^{\infty} a_{n}=5\).

A company buys a machine for \(\$ 225,000\) that depreciates at a rate of \(30 \%\) per year. Find a formula for the value of the machine after \(n\) years. What is its value after 5 years?

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.

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 \(\lim _{n \rightarrow \infty} a_{n}\).

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.

In an experiment, three people toss a fair coin one at a time until one of them tosses a head. Determine, for each person, the probability that he or she tosses the first head. Verify that the sum of the three probabilities is 1 .