Chapter 11

Express the number as a ratio of intergers. \( 5. \overline {71358} \)

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \frac {(-3)^n}{n!} \)

Use series to approximate the definite integral to within the indicated accuracy. \( \int^{1/2}_0 x^3 \arctan x dx \) \( \text { (four decimal places)} \)

Find the values of \( x \) for which the series converges. Find the sum of the series for those values of \( x. \) \( \displaystyle \sum_{n = 1}^{\infty} (-5)^n x^n \)

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequence .) \( a_n = (-1)^n \frac {n}{n + 1} \)

Use series to approximate the definite integral to within the indicated accuracy. \( \int^1_0 \sin (x^4) dx \) \( \text { (four decimal places)} \)

Find the values of \( x \) for which the series converges. Find the sum of the series for those values of \( x. \) \( \displaystyle \sum_{n = 1}^{\infty} (x + 2)^n \)

Find the values of \( x \) for which the series converges. Find the sum of the series for those values of \( x. \) \( \displaystyle \sum_{n = 0}^{\infty} \frac {(x - 2)^n}{3^n} \)

Find the values of \( x \) for which the series converges. Find the sum of the series for those values of \( x. \) \( \displaystyle \sum_{n = 0}^{\infty} (-4)^n (x - 5)^n \)

Use series to approximate the definite integral to within the indicated accuracy. \( \int^{0.5}_0 x^2 e^{-x^{2}} dx \) \( \left( \mid \text {error} \mid < 0.001 \right) \)

Use series to evaluate the limit. \( \lim_{x \to 0} \frac {x - \ln (1 + x)}{ x^2} \)

Find the values of \( x \) for which the series converges. Find the sum of the series for those values of \( x. \) \( \displaystyle \sum_{n = 0}^{\infty} \frac {2^n}{x^n} \)

Use series to evaluate the limit. \( \lim_{x \to 0} \frac {1 - \cos x}{1 + x - e^x} \)

Find the values of \( x \) for which the series converges. Find the sum of the series for those values of \( x. \) \( \displaystyle \sum_{n = 0}^{\infty} \frac {\sin^n x}{3^n} \)

Use series to evaluate the limit. \( \lim_{x \to 0} \frac {\sin x - x + \frac {1}{6} x^3}{x^5} \)

Find the values of \( x \) for which the series converges. Find the sum of the series for those values of \( x. \) \( \displaystyle \sum_{n = 0}^{\infty} e ^{nx} \)

Use series to evaluate the limit. \( \lim_{x \to 0} \frac {\sqrt {1 + x} - 1 - \frac {1}{2} x }{x^2} \)

We have seen that the harmonic series is a divergent series whose terms approach 0. Show that \( \displaystyle \sum_{n = 1}^{\infty} \ln \left( 1 + \frac {1}{n} \right) \) is another series with this property.

(a) Determine whether the sequence defined as follows is convergent or divergent: \( a_1 = 1 \) \( a_{n + 1} = 4 - a_n \) for \( n \ge 1 \) (b) What happens if the first term is \( a_1 = 2 \) ?

Use series to evaluate the limit. \( \lim_{x \to 0} \frac {x^3 - 3x + 3 \tan^{-1} x}{x^5} \)

If \( \$ \)1000 is invested at \( 6 \% \) interest, compounded annually, then after \( n \) years the investment is worth \( a_n = 1000(1.06)^n \) dollars. (a) Find the first five terms of the sequence \( \\{ a_n\\}. \) (b) Is the sequence convergent or divergent? Explain.

Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using CAS to sum the series directly. \( \displaystyle \sum_{n = 1}^{\infty} \frac {3n^2 + 3n + 1}{(n^2 + n)^3} \)

Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using CAS to sum the series directly. \( \displaystyle \sum_{n = 3}^{\infty} \frac {1}{n^5 - 5n^3 + 4n} \)

If you deposit \( \$ \)100 at the end of every month into an account that pays \( 3 \% \) interest per year compounded monthly, the amount of interest accumulated after \( n \) months is given by the sequence \( I_n = 100 \left( \frac {1.0025^n - 1}{0.0025} - n\right) \) (a) Find the first six terms of the sequence. (b) How much interest will you have earned after two years?

Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function. \( y = e^{-x^2} \cos x \)

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 \( \sum_{n = 1}^{\infty} a_n. \)

A fish has 5000 catfish in his pond. The number of catfish by \( 8 \% \) per month and the farmer harvests 300 catfish per month. (a) Show that the catfish population \( P_n \) after \( n \) months is given recursively by \( P_n = 1.08 P_{n-1} - 300\) \(P_0 = 5000\) (b) How many catfish are in the pond after six months?

Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function. \( y = \sec x \)

If the \( n \)th partial sum of a series \( \sum_{n = 1}^{\infty} a_n \) is \( s_n = 3 - n2^{-n}, \) find \( a_n \) and \( \sum_{n = 1}^{\infty} a_n. \)

Find the first 40 terms of the sequence defined by \( a_{n + 1} =\left\\{ \begin{array}{ll} \frac{1}{2} a_n & \text{if } a_n \text{ is an even number} \\ 3a_n + 1 & \text{if } a_n \text{ is an odd number } \end{array} \right. \) and \( a_1 = 11. \) Do the same if \( a_1 = 25. \) Make a conjecture about this type of sequence.

Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function. \( y = \frac {x}{\sin x} \)

A doctor prescribes a 100-mg antibiotic tablet to be taken every eight hours. Just before each tablet is taken, \( 20% \) of the drug remains in the body. (a) How much of the drug is in the body just after the second tablet is taken? After the third tablet? (b) If \( Q_n \) is the quantity of the antibiotic in the body just after the \( n \)th tablet is taken, find an equation that expresses \( Q_{n + 1} \) in terms of \( Q_n. \) (c) What quantity of the antibiotic remains in the body in the long run?

For what values of \( r \) is the sequence \( \left\\{ nr^n \right\\} \) convergent?

Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function. \( y = e^x \ln (1 + x) \)

A patient is injected with a drug every 12 hours. Immediately before each injection the concentration of the drug has been reduced by \( 90\% \) and the new dose in increase the concentration by 1.5 mg/L. (a) What is the concentration after three doses? (b) If \( C_n \) is the concentration after the \( n \)th dose, find a formula for \( C_n \) as a function of \( n. \) (c) What is the limiting value of the concentration?

(a) If \( \left \\{ a_n \right\\} \) is convergent, show that \( \displaystyle\lim_{n\to\infty} a_{n+1} = \displaystyle\lim_{n\to\infty} a_n \) (b) A sequence \( \left\\{ a_n \right\\} \) is defined by \( a_1 = 1 \) and \( a_{n + 1} = 1/(1 + a_n) \) for \( n \ge 1. \) Assuming that \( \left\\{ a_n \right\\} \) is convergent, find its limit.

Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function. \( y = (\arctan x)^2 \)

A patient takes 150 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 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?

Suppose you know that \( \left\\{ a_n \right\\} \) is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can you say about the value of the limit?

Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function. \( y = e^x \sin^2 x \)

After injection of a does \( D \) of insulin, the concentration of insulin in patient's system decays exponentially and so it can be written as \( De^{-at}, \) 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 \( T. \)

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = \cos n \)

Find the sum of the series. \( \sum_{n = 0}^{\infty} (-1)^n \frac {x^{4n}}{n!} \)

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 \( 100c% \) and saves \( 100s% \) 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 \( \lim_{n \to \infty} S_n = kD, \) 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 lending a large percentage of the money that they receive in deposits.

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = \frac{1}{2n + 3} \)

Find the sum of the series. \( \sum_{n = 0}^{\infty} \frac {(-1)^n \pi^{2n}}{6^{2n}(2n)!} \)

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 \( rh, \) where \( 0 < r < 1\. \) Suppose that the ball is dropped from an initial height of \( H \) meters. (a) Assuming that the ball continues to bounce indefinitely, find the total distance that it travels. (b) Calculate the total time that the ball travels. (Use the fact that the ball falls \( \frac {1}{2} gt^2 \) meters in \( t \) seconds.) (c) Suppose that each time the ball strikes the surface with velocity \( v \) it rebounds with velocity \( -kv, \) where \( 0 < k < 1\. \) How long will it take for the ball to come to rest?

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? \( a_n = \frac{1 - n}{2 +n} \)

Find the sum of the series. \( \sum_{n = 1}^{\infty} (-1)^{n-1} \frac {3^n}{n 5^n} \)

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