1.99
See Answer

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

x? dx V1 + x3

** >** List the first five terms of the sequence.
а, — 2, ал н %3D 1+ а,

** >** List the first five terms of the sequence.
an aj = 6, ant1 n

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
2 + n 11 – 2n

** >** A dog sees a rabbit running in a straight line across an open field and gives chase. In a rectangular coordinate system (as shown in the figure), assume: (i) The rabbit is at the origin and the dog is at the point sL, 0d at the instant the dog first sees

** >** Evaluate the integral. /

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
1 2 + 81 2 27 243 729

** >** List the first five terms of the sequence.
а, — 1, а,м 1, an+1 5а, — 3

** >** List the first five terms of the sequence.
(-1)"n an п! + 1

** >** (a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.

** >** Test the series for convergence or divergence.
sin 2n n-1 1+ 2" WI

** >** (a) What is a sequence? (b) What does it mean to say that / (c) What does it mean to say that /

** >** In the figure at the right there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the cir

** >** (a) A sequence {an} is defined recursively by the equation an = Â½ (an-1 + an-2) for n â‰¥ 3, where a1 and a2 can be any real numbers. Experiment with various values of a1 and a2 and use your calculator to guess the limit of t

** >** The Cantor set, named after the German mathematician Georg Cantor (1845â€“1918), is constructed as follows. We start with the closed interval [0, 1] and remove the open interval /.That leaves the two intervals / and we remove the open mid

** >** Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
( sin?a da Jo

** >** Use a Maclaurin series to obtain the Maclaurin series for the given function.
f(x) = 4 + x2

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
1 1 1 3 6 12 15 + + +

** >** The Fibonacci sequence was defined in SectionÂ 11.1 by the equations Show that each of the following statements is true.
fi = 1, f= 1, fn = fa=1 + fa-2 n> 3 %3D 1 1 1 (a) fn-1 far1 fa-1 fn fn fa+1 1 fa (b) Σ 1 ( cC) Σ 2 -2 Jn-1 fa+1 -2 fn-

** >** Snow began to fall during the morning of February 2 and continued steadily into the afternoon. At noon a snowplow began removing snow from a road at a constant rate. The plow traveled 6 km from noon to 1 pm but only 3 km from 1 pm to 2 pm. When did the s

** >** Test the series for convergence or divergence.
3" n? n!

** >** Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
e dx .2

** >** What is wrong with the following calculation?
0 = 0 + 0 + 0 + ... = (1 – 1) + (1 – 1) + (1 – 1) + ... =1 – 1 + 1 – 1 + 1 – 1 + · .. =1 + (-1+ 1) + (-1+ 1) + (-1 + 1) + ... = 1 + 0 + 0 + 0 + ...= 1

** >** Determine whether the geometric series is convergent or divergent. If it is convergent, find its
6 · 2 2n-1 Σ 3"

** >** Determine whether the geometric series is convergent or divergent. If it is convergent, find its
3*+1 Σ (-2)"

** >** The figure shows two circles C and D of radius 1 that touch at P. The line T is a common tangent line; C1 is the circle that touches C, D, and T; C2 is the circle that touches C, D, and C1; C3 is the circle that touches C, D, and C2. This procedure can b

** >** Graph the curves y = xn, 0
1 = 1 п(п + 1) n=1

** >** A peach pie is taken out of the oven at 5:00 pm. At that time it is piping hot, 100 8C. At 5:10 pm its temperature is 80 8C; at 5:20 pm it is 65 8C. What is the temperature of the room?

** >** In Example 9 we showed that the harmonic series is divergent. Here we outline another method, making use of the fact that / If sn is the nth partial sum of the harmonic series, show that / Why does this imply that the harmonic series is divergent?

** >** Find the value of c such that
Σ e10

** >** Find the value of c if
Σ (1 + )"-2 n-2

** >** Evaluate the integral.
'w/3 sin x In(cos x) dx Jo

** >** 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

** >** 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 hypotheti

** >** Test the series for convergence or divergence.
1 Σ k-i k/k2 + 1

** >** Determine whether the geometric series is convergent or divergent. If it is convergent, find its
2n 6*-1

** >** 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 De-at, where t represents time in hours and a is a positive constant. (a) If a dose D is injected every T hours, w

** >** Use Poiseuille’s Law to calculate the rate of flow in a small human artery where we can take / /

** >** 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 after the third tablet? After the nth tablet? (b) What quantity of the drug re

** >** 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 increases the concentration by 1.5 mg/L. (a) What is the concentration after three doses? (b) If C

** >** 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

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
cos'n an 2"

** >** 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 the CAS to sum the series directly.
1 Σ n° – 5n° + 4n n-3 N

** >** 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 the CAS to sum the series directly.
Зп? + Зп + 1 Σ (n² + n)³ n-1

** >** Test the series for convergence or divergence.
1 n/In n n-2

** >** Determine whether the geometric series is convergent or divergent. If it is convergent, find its
Σ 12(0.73 )" 1

** >** Suppose that f(1) = 2, f(4) = 7, f9(1) = 5, f9(4) = 3, and f 0 is continuous. Find the value of
fxf"(x) dx.

** >** Test the series for convergence or divergence.
π Σ (-1)" sin |

** >** Test the series for convergence or divergence.
n cos nT 2"

** >** Test the series for convergence or divergence.
sin(n + )m Σ 1 + Jn n-0 8

** >** Test the series for convergence or divergence.
Σ(-1)" ' arctan n

** >** Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.
(-1)" Σ =2 In n

** >** Test the series for convergence or divergence.
E-le?/n , 2/m n-1

** >** Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.
y = x – In x, 1<x<4

** >** Test the series for convergence or divergence.
E (-1)"+'ne

** >** We have seen that the harmonic series is a divergent series whose terms approach 0. Show that is another series with this property.
E In 1 +

** >** Test the series for convergence or divergence.
n? E(-1)**1. n3 + 4 n-1

** >** Test the series for convergence or divergence.
E (-1)" 2n + 3 n-1

** >** Test the series for convergence or divergence.
E (-1)"e ™

** >** If f(0) = t(0) = 0 and f 0 and t0 are continuous, show that
"S9g"(x) dx = f(a)g\(a) – f'(a)g(a) + [' r"(x)g(x) dx %3D

** >** Use a Maclaurin series to obtain the Maclaurin series for the given function.
f(x) V2 + x

** >** Test the series for convergence or divergence.
n? E (-1)": n' п? + п +1 n-1

** >** Test the series for convergence or divergence.
Зп — 1 2 (-1)", 2n + 1

** >** Test the series for convergence or divergence.
(-1)**1 Σ Vn + 1 n-0

** >** Test the series for convergence or divergence.
(-1)* -1 Σ 3 + 5n

** >** Test the series for convergence or divergence.
1 1 1 1 + In 6 1 In 3 In 4 In 5 In 7

** >** Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.
y = sin x, 0 <I<T

** >** Find the values of x for which the series converges. Find the sum of the series for those values of x.
00 Σ E e" n-0

** >** Use the Root Test to determine whether the series is convergent or divergent.
Σ 1 + 8

** >** Test the series for convergence or divergence.
-중 +8-우 + #-4 +

** >** Test the series for convergence or divergence.
를-3+ 3-3 + 류-.

** >** Prove part (i) of Theorem 8.

** >** (a) What is an alternating series? (b) Under what conditions does an alternating series converge? (c) If these conditions are satisfied, what can you say about the remainder after n terms?

** >** Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
x? + 4 A can be put in the form В x*(x – 4) х — 4

** >** A particle that moves along a straight line has velocity v(t) − t2e2t meters per second after t seconds. How far will it travel during the first t seconds?

** >** Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
x? + 4 A can be put in the form B C x(x² – 4) x + 2 x - 2

** >** Use Simpson’s Rule with n = 6 to estimate the area under the curve y = ex/x from x = 1 to x = 4.

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
4 + 3 + + + 16

** >** Use the Table of Integrals on Reference Pages 6â€“10 to evaluate the integral.
cos x dx sin?x – 9

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Use integration by parts to prove the reduction formula.
x"e*dx = x"e* – n |x" 'e*dx

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Use the Root Test to determine whether the series is convergent or divergent.
5n -2n Σ n + 1, n-

** >** Evaluate the integral.
- tan 0 1 + tan 0

** >** Express the repeating decimal 4.17326326326... as a fraction.

** >** A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is m, the fuel is consumed at rate r, and the exhaust gases are ejected with constant velocity ve (r

** >** Evaluate the integral.
1* In t dt

** >** Find the sum of the series.
e? 1-е + 2! 3! 4! +

** >** Test the series for convergence or divergence.
Σ 3"

** >** Find the sum of the series.
(-1)" 7" 32"(2n)!

** >** Express the number as a ratio of integers.
0.46 = 0.46464646 . . .

** >** Determine whether the sequence converges or diverges. If it converges, find the limit
2 3 + 5n? 1 +n

** >** In Section 4.8 we considered Newtonâ€™s method for approximating a root r of the equation f(x) = 0, and from an initial approximation x1 we obtained successive approximations x2, x3, . . . , where Use Taylorâ€™s Inequality

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
In n In 2n

** >** Evaluate the integral.
dx J31+ |x|

** >** Evaluate the integral.
/2 cos'x sin 2x dx

** >** Determine whether the series is conditionally convergent, absolutely convergent, or divergent.
E(-1)" 'n-3

** >** Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter.
x = t³ + 6t + 1, y=2t – t2; t=-1

** >** Find a polar equation for the curve represented by the given Cartesian equation. x2 + y2 = 2

** >** Use the Root Test to determine whether the series is convergent or divergent.
(-1)“ 1 Σ (In n)" R-2