Question: Use the binomial series to expand the
Use the binomial series to expand the function as a power series. State the radius of convergence.
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1 (2 + x)
> A sequence {an} is defined recursively by the equations Find the sum of the series / dо — ај — 1 п(n — 1)а, — (п - 1)(п — 2)а, 1 — (п — 3)а, 2
> Suppose that circles of equal diameter are packed tightly in n rows inside an equilateral triangle. (The figure illustrates the case n = 4.) If A is the area of the triangle and An is the total area occupied by the n rows of circles, show that An lim
> If p > 1, evaluate the expression 1 + 2" 1 + 4P 3P 1 + 4P 1 1 2" 3P + + +
> Find the sum of the series /
> Suppose you have a large supply of books, all the same size, and you stack them at the edge of a table, with each book extending farther beyond the edge of the table than the one beneath it. Show that it is possible to do this so that the top book extend
> Find the interval of convergence of /
> If a0 + a1 + a2 + ∙∙∙ + ak = 0, show that If you don’t see how to prove this, try the problem-solving strategy of using analogy (see page 71). Try the special cases k = 1 and k = 2
> A car is moving with speed 20 m/s and acceleration 2 m/s2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled duri
> Use the result of Problem 7(a) to find the sum of the series / Data from Problem 7: (a) Show that for xy ≠2, if the left side lies between / (b) Show that / (c) Deduce the following formula of John Machin (1680–1751)
> (a) Prove a formula similar to the one in Problem 7(a) but involving arccot instead of arctan. (b) Find the sum of the series / Data from Problem 7: (a) Show that for xy ≠2, if the left side lies between / (b) Show that / (c) Deduce t
> (a) Show that for xy ≠2, if the left side lies between / (b) Show that / (c) Deduce the following formula of John Machin (1680–1751): (d) Use the Maclaurin series for arctan to show that (e) Show that (f) Deduce that,
> Find the sum of the series where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s. 1 + 6 9 12 + - l00 + | + + - l3
> Evaluate the integral. 1/2 x cos TX dx Jo
> Evaluate the integral. |(x – 1) sin 7x dx
> To construct the snowflake curve, start with an equilateral triangle with sides of length 1. Step 1 in the construction is to divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part
> Not all water tanks are shaped like cylinders. Suppose a tank has cross-sectional area A(h) at height h. Then the volume of water up to height / and so the Fundamental Theorem of Calculus gives dV/dh = A(h). It follows that and so Torricelliâ€
> In many parts of the world, the water for sprinkler systems in large hotels and hospitals is supplied by gravity from cylindrical tanks on or near the roofs of the buildings. Suppose such a tank has radius 10 ft and the diameter of the outlet is 2.5 inch
> Because of the rotation and viscosity of the liquid, the theoretical model given by Equation 1 isn’t quite accurate. Instead, the model is often used and the constant k (which depends on the physical properties of the liquid
> (a) Suppose the tank is cylindrical with height 6 ft and radius 2 ft and the hole is circular with radius 1 inch. If we take t = 32 ft/s2, show that h satisfies the differential equation (b) Solve this equation to find the height of the water at time t,
> Find the sum of the series. 1 3. 23 1 1 1 1. 2 '5 · 25 7. 27
> Suppose you know that and the Taylor series of f centered at 4 converges to f(x) for all x in the interval of convergence. Show that the fifth degree Taylor polynomial approximates f(5) with error less than 0.0002. (-1)"n! f®(4) 3"(л + 1)
> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. f(x) = a =
> Find the sum of the series. 27 + 4! 81 3 + 2! 3! +
> Find the sum of the series. (In 2)? 1 - In 2 + 2! (In 2) 3!
> Evaluate the integral. | (arcsin x)? dx
> Find the sum of the series. (-1)"72 +1 Σ 42m(2n + 1)! 2n+1,
> Find the sum of the series. 3" Σ no 5"n!
> Find the sum of the series. 3" E (-1)" -1. n 5"
> Find the sum of the series. (-1)"72" 6ª"(2n)!
> Find the sum of the series. An E(-1)": n! -0
> The terms of a series are defined recursively by the equations Determine whether / converges or diverges. 5n + 1 an 4n + 3 а, — 2 ant1
> Use a Maclaurin series to obtain the Maclaurin series for the given function. S(x) = e³* – e2*
> Use the Alternating Series Estimation Theorem or Taylor’s Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. arctan x = x - 3 (ler
> Test the series for convergence or divergence. E (v2 – 1)" 11
> Evaluate the integral. xe 2x (1 + 2x)?
> Use a Maclaurin series to obtain the Maclaurin series for the given function. S(a) — sin(тx/4)
> Use a Maclaurin series to obtain the Maclaurin series for the given function. f(x) = arctan(x²)
> Use the Root Test to determine whether the series is convergent or divergent. Σ (arctan n"
> Use the Root Test to determine whether the series is convergent or divergent. (-2)" Σ n"
> Use the Ratio Test to determine whether the series is convergent or divergent. n! 100"
> Find the radius of convergence and interval of convergence of the series. 2 n"x" -1
> Test the series for convergence or divergence. e 2 一1 1
> Use a Maclaurin series to obtain the Maclaurin series for the given function. S(x)- = x cos(}x²)
> Use the Alternating Series Estimation Theorem or Taylor’s Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. cos x = 1 2 (lerror|<
> Evaluate the integral. x tan?x dx
> Use a Maclaurin series to obtain the Maclaurin series for the given function. f(x) x cos 2x
> The Bessel function of order 1 is defined by (a) Show that J1 satisfies the differential equation (b) Show that / (-1)"x²*+1 J,(x) = E n! (n + 1)!2²n*! x*I"(x) + xJ{(x) + (x² – 1)J,(x) = 0
> (a) Show that J0 (the Bessel function of order 0 given in Example 4) satisfies the differential equation (b) Evaluate / correct to three decimal places. X²JF(x) + x.JK(x) + x³J(x) = 0
> Use the binomial series to expand the function as a power series. State the radius of convergence. (1 – x)/4
> Use the binomial series to expand the function as a power series. State the radius of convergence. V8 + x
> Use the binomial series to expand the function as a power series. State the radius of convergence. V1 - x
> Prove that the series obtained in Exercise 18 represents cosh x for all x. Data from Exercise 18: Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) â
> Prove that the series obtained in Exercise 17 represents sinh x for all x. Data from Exercise 17: Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) â
> Evaluate the integral. z³e² dz
> Use the Alternating Series Estimation Theorem or Taylor’s Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. sin x = x - (lerror|
> Prove that the series obtained in Exercise 25 represents sin x for all x. Data from Exercise 25: Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0
> Prove that the series obtained in Exercise 13 represents cos x for all x. Data from Exercise 13: Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) â&
> Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. S(x) = Vx, a = 16
> Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) = sin x, a = T а —
> Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) = cos x, a= /2 a = "/2
> Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) — е", а — 3 %3D
> Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) — 1/x, а —-3 a
> Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) — In x, а — 2
> Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) = x — х* — х* + 2, а—-2
> Evaluate the integral. Se°cos 20 do
> Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) = x* + 2x³ + x, a = 2
> How many terms of the Maclaurin series for ln(1 + x) do you need to use to estimate ln 1.4 to within 0.001?
> Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) = cosh x
> Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) = sinh x
> Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) - x cos x
> Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. S(x) = 2"
> Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) = e-24
> Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) = cos x
> Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. f(x) = In(1 + x)
> Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. -2 f(x) = (1 – x) ?
> Evaluate the integral. e 20 sin 30 d0
> Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. f(x) — сos?x, а — 0 = cos
> Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. f(x) — sin x, а — п/6
> Use Taylor’s Inequality to determine the number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to within 0.00001.
> Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. f(x) — In x, а — 1
> Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. f(x) — Vх, а —8
> Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. 1 S(x) a = 2 1 + x'
> Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a.
> Find the Taylor series for f centered at 4 if What is the radius of convergence of the Taylor series? (-1)" n! f(4) 3"(n + 1)
> If f (n)(0) = ( n+1)! for n = 0, 1, 2, ……, find the Maclaurin series for f and its radius of convergence.
> The graph off is shown. (a) Explain why the series is not the Taylor series of f centered at 1. (b) Explain why the series is not the Taylor series of f centered at 2. y f 1 1.6 – 0.8(x – 1) + 0.4(x – 1)? – 0.1(x – 1)³ + ... 2.8 + 0.5(x – 2) + 1.5(
> Evaluate the integral. dz 107
> Test the series for convergence or divergence. п — 1 n-1 n° +1
> Test the series for convergence or divergence. n? – 1 -1 n° + 1
> Use the information from Exercise 16 to estimate sin 38° correct to five decimal places. Data from Exercise 16: (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor’s Inequality to estimate
> Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. (-1)" –1)" -2 п In n
> Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. sin(nT/6) 1 + n/n R-1
> Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. Σ -2In n
> Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. n52n 10**1 n+1
> Use the Ratio Test to determine whether the series is convergent or divergent. 2.4· 6. .... (2n) Σ n!
> Use the Ratio Test to determine whether the series is convergent or divergent. 2.5· 8· 11 3. 5·7.9 2 2·5 2 ·5- 8 3 3. 5 3.5.7 +
> Evaluate the integral. | (In x)°dx
> Use the Ratio Test to determine whether the series is convergent or divergent. 2! 3! 4! 1:3 1.3. 5 1:3. 5·7 п! + (-1)"-1. + 1. 3· 5. .... (2n – 1)
> Use the Ratio Test to determine whether the series is convergent or divergent. (2n)! Σ (л!)? n=1
> Use the Ratio Test to determine whether the series is convergent or divergent. n 100 100" Σ n!
