Problem 67
[T] Find the first 1000 digits of \(\pi\) using either a computer program or Internet resource. Create a bit sequence \(b_{n}\) by letting \(b_{n}=1\) if the \(n\) th digit of \(\pi\) is odd and \(b_{n}=0\) if the \(n\) th digit of \(\pi\) is even. Compute the average value of \(b_{n}\) and the average value of \(d_{n}=\left|b_{n+1}-b_{n}\right|, n=1, \ldots, 999 .\) Does the sequence \(b_{n}\) appear random? Do the differences between successive elements of \(b_{n}\) appear random?
Problem 68
The following alternating series converge to given multiples of \(\pi .\) Find
the value of \(N\) predicted by the remainder estimate such that the Nth partial
sum of the series accurately approximates the left-hand side to within the
given error. Find the minimum \(N\) for which the error bound holds, and give
the desired approximate value in each case. Up to 15 decimals places,
\(\pi=3.141592653589793 .\)[T] The series \(\sum_{n=0}^{\infty} \frac{\sin (x+\pi
n)}{x+\pi n}\) plays an important role in signal processing. Show that
\(\sum_{n=0}^{\infty} \frac{\sin (x+\pi n)}{x+\pi n}\) converges whenever
\(0
Problem 68
The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if \(\lim _{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\), then \(\sum a_{n}\) converges, while if \(\lim _{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\), then \(\sum a_{n}\) diverges. Let \(a_{n}=\frac{1}{4} \frac{3}{6} \frac{5}{8} \cdots \frac{2 n-1}{2 n+2}=\frac{1 \cdot 3-5 \cdots(2 n-1)}{2^{n}(n+1) !} .\) Explain why the ratio test cannot determine convergence of \(\sum_{n=1}^{\infty} a_{n} .\) Use the fact that \(1-1 /(4 k)\) is increasing \(k\) to estimate \(\lim _{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}\).
Problem 69
The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if \(\lim _{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\), then \(\sum a_{n}\) converges, while if \(\lim _{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\), then \(\sum a_{n}\) diverges. Let \(a_{n}=\frac{1}{1+x} \frac{2}{2+x} \cdots \frac{n}{n+x} \frac{1}{n}=\frac{(n-1) !}{(1+x)(2+x) \cdots(n+x)} .\) Show that \(a_{2 n} / a_{n} \leq e^{-x / 2} / 2 .\) For which \(x>0\) does the generalized ratio test imply convergence of \(\sum_{n=1}^{\infty} a_{n} ?\) (Hint: Write \(2 a_{2 n} / a_{n}\) as a product of \(n\) factors each smaller than \(1 /(1+x /(2 n))\)
Problem 70
The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if \(\lim _{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\), then \(\sum a_{n}\) converges, while if \(\lim _{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\), then \(\sum a_{n}\) diverges. Let \(a_{n}=\frac{\pi^{\ln n}}{(\ln n)^{n}} .\) Show that \(\frac{a_{2 n}}{a_{n}} \rightarrow 0\) as \(n \rightarrow \infty\).
Problem 71
True or False? Justify your answer with a proof or a counterexample. $$ \text { If } \lim _{n \rightarrow \infty} a_{n}=0, \text { then } \sum_{n=1}^{\infty} a_{n} \text { converges } $$
Problem 72
True or False? Justify your answer with a proof or a counterexample. $$ \text { If } \lim _{n \rightarrow \infty} a_{n} \neq 0 \text { , then } \sum_{n=1}^{\infty} a_{n} \text { diverges. } $$
Problem 73
True or False? Justify your answer with a proof or a counterexample. $$ \text { If } \sum_{n=1}^{\infty}\left|a_{n}\right| \text { converges, then } \sum_{n=1}^{\infty} a_{n} \text { converges. } $$
Problem 74
The following alternating series converge to given multiples of \(\pi .\) Find the value of \(N\) predicted by the remainder estimate such that the Nth partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(N\) for which the error bound holds, and give the desired approximate value in each case. Up to 15 decimals places, \(\pi=3.141592653589793 .\)[T] The alternating harmonic series converges because of cancellation among its terms. Its sum is known because the cancellation can be described explicitly. A random harmonic series is one of the form \(\sum_{n=1}^{\infty} \frac{S_{n}}{n}\), where \(s_{n}\) is a randomly generated sequence of \(\pm 1\) 's in which the values \(\pm 1\) are equally likely to occur. Use a random number generator to produce 1000 random \(\pm 1\) s and plot the partial sums \(S_{N}=\sum_{n=1}^{N} \frac{s_{n}}{n}\) of your random harmonic sequence for \(N=1\) to \(1000 .\) Compare to a plot of the first 1000 partial sums of the harmonic series.
Problem 75
Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit. $$ a_{n}=\frac{3+n^{2}}{1-n} $$
