Chapter 8

Let $$ f(x)=\sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} $$ Find the intervals of convergence for \(f, f^{\prime},\) and \(f^{\prime \prime}\)

If the \(n\) th partial sum of a series \(\sum_{n=1}^{\infty} a_{n}\) is $$s_{n}=\frac{n-1}{n+1}$$ find \(a_{n}\) and \(\sum_{n=1}^{\infty} a_{n}\)

The meaning of the decimal representation of a number $$0 . d_{1} d_{2} d_{3} \ldots\( (where the digit \)d_{i}\( is one of the numbers \)0,1\( \)2, \ldots, 9 )\( is that \)\quad 0 . d_{1} d_{2} d_{3} d_{4} \ldots=\frac{d_{1}}{10}+\frac{d_{2}}{10^{2}}+\frac{d_{3}}{10^{3}}+\frac{d_{4}}{10^{4}}+\cdots$$ Show that this series always converges.

Use the Maclaurin series for cos \(x\) to compute \(\cos 5^{\circ}\) correct to five decimal places.

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot(2 n)}{n !} $$

\(37-40=\) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=n(-1)^{n}$$

(a) Starting with the geometric series \(\sum_{n=0}^{\infty} x^{n},\) find the sum of the series $$ \sum_{n=1}^{\infty} n x^{n-1} \quad|x|<1 $$ (b) Find the sum of each of the following series. $$ \text { (i) }\sum_{n=1}^{\infty} n x^{n}, \quad|x|<1 \quad \text { (ii) } \sum_{n=1}^{\infty} \frac{n}{2^{n}} $$ (c) Find the sum of each of the following series. $$ \text { (i) }\sum_{n=2}^{\infty} n(n-1) x^{n}, \quad|x|<1 $$ $$ \text { (ii) }\sum_{n=2}^{\infty} \frac{n^{2}-n}{2^{n}} \quad \text { (iii) } \sum_{n=1}^{\infty} \frac{n^{2}}{2^{n}} $$

If the \(n\) th partial sum of a series \(\Sigma_{n=1}^{\infty} a_{n}\) is \(s_{n}=3-n 2^{-n}\) find \(a_{n}\) and \(\sum_{n=1}^{\infty} a_{n}\)

Show that if \(a_{n}>0\) and \(\Sigma a_{n}\) is convergent, then \(\Sigma \ln \left(1+a_{n}\right)\) is convergent.

Use the Maclaurin series for \(e^{x}\) to calculate 1\(/ \sqrt[10]{e}\) correct to five decimal places.

\(19-40=\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{2^{n} n !}{5 \cdot 8 \cdot 11 \cdot \cdots \cdot(3 n+2)} $$

\(37-40=\) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=n+\frac{1}{n}$$

Use the power series for tan \(^{-1} x\) to prove the following expression for \(\pi\) as the sum of an infinite series: $$ \pi=2 \sqrt{3} \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2 n+1) 3^{n}} $$

A patient takes 150 \(\mathrm{mg}\) of a drug at the same time every day. Just before each tablet is taken, 5\(\%\) of the drug remains in the body. (a) What quantity of the drug is in the body after the third tablet? After the \(n\) th tablet? (b) What quantity of the drug remains in the body in the long run?

If \(\Sigma a_{n}\) is a convergent series with positive terms, is it true that \(\sum \sin \left(a_{n}\right)\) is also convergent?

\(41-42=\) Let \(\left\\{b_{n}\right\\}\) be a sequence of positive numbers that con- verges to \(\frac{1}{2} .\) Determine whether the given series is absolutely convergent. $$\sum_{n=1}^{\infty} \frac{b_{n}^{n} \cos n \pi}{n}$$

\(\begin{array}{l}{\text { (a) Use the binomial series to expand } 1 / \sqrt{1-x^{2}}}. \\ {\text { (b) Use part (a) to find the Maclaurin series for } \sin ^{-1} x}.\end{array}\)

Find the limit of the sequence $$\\{\sqrt{2}, \sqrt{2 \sqrt{2}}, \sqrt{2 \sqrt{2 \sqrt{2}}}, \ldots\\}$$

Find the sum of the series $$ \sum_{n=1}^{\infty} \frac{4^{n}}{n 5^{n}} $$

After injection of a dose \(D\) of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as \(D e^{-a t}\) , where \(t\) represents time in hours and \(a\) is a positive constant. (a) If a dose \(D\) is injected every \(T\) hours, write an expression for the sum of the residual concentrations just before the \((n+1)\) st injection. (b) Determine the limiting pre-injection concentration. (c) If the concentration of insulin must always remain at or above a critical value \(C,\) determine a minimal dosage \(D\) in terms of \(C, a,\) and \(T\)

(a) Suppose that \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms and \(\Sigma b_{n}\) is convergent. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=0\) then \(\Sigma a_{n}\) is also convergent. (b) Use part (a) to show that the series converges. $$\sum_{n=1}^{\infty} \frac{\ln n}{n^{3}} \quad \text { (ii) } \sum_{n=1}^{\infty} \frac{\ln n}{\sqrt{n} e^{n}} $$

\(41-42=\) Let \(\left\\{b_{n}\right\\}\) be a sequence of positive numbers that con- verges to \(\frac{1}{2} .\) Determine whether the given series is absolutely convergent. $$\sum_{n=1}^{\infty} \frac{(-1)^{n} n !}{n^{n} b_{1} b_{2} b_{3} \cdots b_{n}}$$

A sequence \(\left\\{a_{n}\right\\}\) is given by \(a_{1}=\sqrt{2}, a_{n+1}=\sqrt{2+a_{n}}\) (a) By induction or otherwise, show that \(\left\\{a_{n}\right\\}\) is increasing and bounded above by \(3 .\) Apply the Monotonic Sequence Theorem to show that lim \(_{n \rightarrow \infty} a_{n}\) exists. (b) Find \(\lim _{n \rightarrow \infty} a_{n} .\)

\(\begin{array}{l}{\text { (a) Expand } 1 / \sqrt[4]{1+x} \text { as a power series. }} \\ {\text { (b) Use part (a) to estimate } 1 / \sqrt[4]{1.1} \text { correct to three }} \\ {\text { decimal places. }}\end{array}\)

When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending \(D\) dollars. Suppose that each recipient of spent money spends 100\(c \%\) and saves 100\(s \%\) of the money that he or she receives. The values \(c\) and \(s\) are called the marginal propensity to consume and the marginal propensity to save and, of course, \(c+s=1\) . (a) Let \(S_{n}\) be the total spending that has been generated after \(n\) transactions. Find an equation for \(S_{n} .\) (b) Show that \(\lim _{n \rightarrow \infty} S_{n}=k D,\) where \(k=1 / s\) . The number \(k\) is called the multiplier. What is the multiplier if the marginal propensity to consume is 80\(\% ?\) Note: The federal government uses this principle to justify deficit spending. Banks use this principle to justify lending a large percentage of the money that they receive in deposits.

(a) Suppose that \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are series with positive terms and \(\Sigma b_{n}\) is divergent. Prove that if \(\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\infty\) then \(\Sigma a_{n}\) is also divergent. (b) Use part (a) to show that the series diverges. \(\quad\left(\) i) \(\sum_{n=2}^{\infty} \frac{1}{\ln n} \quad\) (ii) \(\sum_{n=1}^{\infty} \frac{\ln n}{n}\right.\)

For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer)? (a) $$\sum_{n=1}^{\infty} \frac{1}{n^{3}}$$ (b) $$\sum_{n=1}^{\infty} \frac{n}{2^{n}}$$ (c) $$\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{\sqrt{n}}$$ (d) $$\sum_{n=1}^{\infty} \frac{\sqrt{n}}{1+n^{2}}$$

Evaluate the indefinite integral as an infinite series. $$\int x \cos \left(x^{3}\right) d x$$

Use induction to show that the sequence defined by \(a_{1}=1\) \(a_{n+1}=3-1 / a_{n}\) is increasing and \(a_{n}<3\) for all \(n .\) Deduce that \(\left\\{a_{n}\right\\}\) is convergent and find its limit.

A certain ball has the property that each time it falls from a height \(h\) onto a hard, level surface, it rebounds to a height \(r h,\) where \(0

For which positive integers \(k\) is the following series convergent? $$\sum_{n=1}^{\infty} \frac{(n !)^{2}}{(k n) !}$$

Evaluate the indefinite integral as an infinite series. $$\int \frac{e^{x}-1}{x} d x$$

Show that the sequence defined by $$a_{1}=2 \quad a_{n+1}=\frac{1}{3-a_{n}}$$ satisfies \(0 < a_{n} \leqslant 2\) and is decreasing. Deduce that the sequence is convergent and find its limit.

Give an example of a pair of series \(\Sigma a_{n}\) and \(\Sigma b_{n}\) with posi- tive terms where \(\lim _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)=0\) and \(\Sigma b_{n}\) diverges, but \(\sum a_{n}\) converges. [Compare with Exercise \(42 . ]\)

Find the value of $$c\( if \)\sum_{n=2}^{\infty}(1+c)^{-n}=2$$

(a) Show that \(\sum_{n=0}^{\infty} x^{n} / n !\) converges for all \(x\). (b) Deduce that \(\lim _{n \rightarrow \infty} x^{n} / n !=0\) for all \(x\).

Evaluate the indefinite integral as an infinite series. $$\int \frac{\cos x-1}{x} d x$$

Find all positive values of \(b\) for which the series \(\sum_{n=1}^{\infty} b^{\ln n}\) converges.

Evaluate the indefinite integral as an infinite series. $$\int \arctan \left(x^{2}\right) d x$$

(a) Let \(a_{1}=a, a_{2}=f(a), a_{3}=f\left(a_{2}\right)=f(f(a)), \ldots\) \(a_{n+1}=f\left(a_{n}\right),\) where \(f\) is a continuous function. If \(\lim _{n \rightarrow \infty} a_{n}=L,\) show that \(f(L)=L\) (b) Illustrate part (a) by taking \(f(x)=\cos x, a=1,\) and estimating the value of \(L\) to five decimal places.

Around \(1910,\) the Indian mathematician Srinivasa Ramanujan discovered the formula $$\frac{1}{\pi}=\frac{2 \sqrt{2}}{9801} \sum_{n=0}^{\infty} \frac{(4 n) !(1103+26390 n)}{(n !)^{4} 396^{4 n}}$$ William Gosper used this series in 1985 to compute the first 17 million digits of \(\pi .\) (a) Verify that the series is convergent. (b) How many correct decimal places of \(\pi\) do you get if you use just the first term of the series? What if you use two terms?

Use series to approximate the definite integral to within the indicated accuracy. $$\int_{0}^{1} x \cos \left(x^{3}\right) d x \quad(\text { three decimal places })$$

Show that if \(a_{n}>0\) and \(\lim _{n \rightarrow \infty} n a_{n} \neq 0,\) then \(\Sigma a_{n}\) is divergent.

Prove the Root Test. [Hint for part (i): Take any number \(r\) such that \(L

We know that \(\lim _{n \rightarrow \infty}(0.8)^{n}=0[\) from \([8]\) with \(r=0.8]\) Use logarithms to determine how large \(n\) has to be so that \((0.8)^{n}<0.000001 .\)

Find all values of \(c\) for which the following series converges. \(\quad \sum_{n=1}^{\infty}\left(\frac{c}{n}-\frac{1}{n+1}\right)\)

Use series to approximate the definite integral to within the indicated accuracy. $$\int_{0}^{1} \sin \left(x^{4}\right) d x \quad(\text { four decimal places })$$

What is wrong with the following calculation? $$\begin{aligned} 0 &=0+0+0+\cdots \\ &=(1-1)+(1-1)+(1-1)+\cdots \\\ &=1-1+1-1+1-1+\cdots \\ &=1+(-1+1)+(-1+1)+(-1+1)+\cdots \\\ &=1+0+0+0+\cdots=1 \end{aligned}$$ (Guido Ubaldus thought that this proved the existence of God because "something has been created out of nothing.")

Use series to approximate the definite integral to within the indicated accuracy. $$\int_{0}^{0.1} \frac{d x}{\sqrt{1+x^{3}}}\left( | \text { error } |<10^{-8}\right)$$

Suppose that \(\sum_{n=1}^{\infty} a_{n}\left(a_{n} \neq 0\right)\) is known to be a convergent series. Prove that \(\sum_{n=1}^{\infty} 1 / a_{n}\) is a divergent series.