(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 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
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x? dx V1 + x3
> 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.
> 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 6–10 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 Newton’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 Taylor’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
> Find a polar equation for the curve represented by the given Cartesian equation. x + y = 2
> Find the values of x for which the series converges. Find the sum of the series for those values of x. sin"x o 3"
> Express the number as a ratio of integers. 0.8 = 0.8888...
> Evaluate the integral. x? + 2 dx x + 2
> Determine whether the series is convergent or divergent. Σ 5"
> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. rV4 - x² dx
> Evaluate the integral. /arctan x dx 1 + x?
> Determine whether the series is convergent or divergent. Σ i n' + 1 R-1
> Find the area of the surface obtained by rotating the curve in Exercise 9 about the y-axis. Data from Exercise 9: Find the length of the curve y - i VJi - I di 1<x< 16
> Evaluate the integral. sin(In t) dt
> Evaluate the integral. dt J 21? + 3t + 1
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. n sin n an n2 + 1
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. Σ
> Find the values of x for which the series converges. Find the sum of the series for those values of x. 2" Σ o x"
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 3. n an 1 + .2 re
> Use the Root Test to determine whether the series is convergent or divergent. п? + 1 2n? + 1
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 9n+1 an 10"
> Calculate the average value of f(x) = x sec2x on the interval /
> Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. (4x? – 3 dx; entry 39
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 2 + n3 an 1+ 2n3
> What does your result from Problem 1 say about the areas A1 and A2 shown in the figure? Data from Problem 1: Suppose the cups have height h, cup A is formed by rotating the curve x = fs/d about the y-axis, and cup B is formed by rotating the same curv
> All solutions of the differential equation y’ = -1 – y4 are decreasing functions.
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 2 arctan n n-1
