Find the sum of the series. ∑∞n=1 (-3)n-1/23n
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> Graph the curve and find the area that it encloses. r= 2 sin e + 3 sin 90
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> (a). Approximate f by a Taylor polynomial with degree n at the number a. (b). Use Taylor’s Inequality to estimate the accuracy of the approximation f (x) ≈ Tn (x) when x lies in the given interval. (c). Check your resu
> 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) = xe хе
> 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) = e5=
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) = tan-'x, a = 1
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) %3D хе -2:, а %3D 0 = xe a = 0
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) In x a = 1
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. = cos x, a = T/2 1 = 7/2
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) = x + e, a=0
> Find the Taylor polynomial T3(x) for the function f at the number a. Graph f and T3 on the same screen. f(x) = 1/x, a = 2
> Graph the curve and find the area that it encloses.
> (a). Find the Taylor polynomials up to degree 3 for f (x) = 1/x centered at a = 1. Graph f and these polynomials on a common screen. (b). Evaluate f and these polynomials at x = 0.9 and 1.3. (c). Comment on how the Taylor polynomials converge to f (x).
> (a). Find the Taylor polynomials up to degree 6 for f (x) = cos x centered at a = 0. Graph f and these polynomials on a common screen. (b). Evaluate f and these polynomials at x = π/4, π/2, and π. (c). Comment on how the Taylor polynomials converge to f
> Use series to evaluate the following limit. limx→0 sin x – x/ x3
> (a). Approximate f by a Taylor polynomial with degree at the number a. (b). Graph f and Tn on a common screen. (c). Use Taylor’s Inequality to estimate the accuracy of the approximation f (X) = Tn(x) when lies in the given interval. (d)
> (a). Approximate f by a Taylor polynomial with degree at the number a. (b). Graph f and Tn on a common screen. (c). Use Taylor’s Inequality to estimate the accuracy of the approximation f (X) = Tn(x) when lies in the given interval. (d)
> Use series to approximate f10√1 + x4, dx correct to two decimal places.
> Evaluate f ex/x dx as an infinite series.
> 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, and tan-1x. f(x)
> 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, and tan-1x. f(x)
> 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, and tan-1x. f(x)
> Sketch the curve and find the area that it encloses. r = 2 + cos 2θ
> 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, and tan-1x. f(x)
> 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, and tan-1x. f(x)
> 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, and tan-1x. f(x)
> 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, and tan-1x. f(x)
> 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, and tan-1x. x? f
> Find the Taylor series of f (x) = cos x at a = π/3.
> Find the Taylor series of f (x) = sin x at a π/6.
> Find the radius of convergence of the series ∑∞n=1 (2n)!/(n!)2 xn.
> Find the radius of convergence and interval of convergence of the series. 2"(х — 3)" Σ In + 3 00
> Find the radius of convergence and interval of convergence of the series. 2"х — 2)" A-1 (п + 2)!
> Sketch the curve and find the area that it encloses. r = 2 cos 3θ
> Find the radius of convergence and interval of convergence of the series. (x + 2)* Σ n4" 00 n-
> Find the radius of convergence and interval of convergence of the series. x' E (-1)" - n25" ,2.
> Prove that if the series ∑∞n=1 an is absolutely convergent, then the series ∑∞n=1 (n + 1/n)an is also absolutely convergent.
> (a). Show that the series ∑∞n=1nn/(2n)! is convergent. (b). Deduce that limn→∞ nn/(2n)! = 0.
> Use the sum of the first eight terms to approximate the sum of the series∑∞n=1 (2 + 5n)-1. Estimate the error involved in this approximation.
> (a). Find the partial sum s5 of the series ∑∞n=1 1/n6 and estimate the error in using it as an approximation to the sum of the series. (b). Find the sum of this series correct to five decimal places.
> Find the sum of the series ∑∞n=1 (-1)n+1 /n5 correct to four decimal places.
> For what values of does the series ∑∞n=1(ln x)n converge?
> Express the repeating decimal as a 1.2345345345… fraction.
> Find the sum of the series. 1 – e + e2/2! + e3/3! + e4/4! - …
> Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 2 <rs 5, 37/4 < 0 < 5T/4
> Find the sum of the series. ∑∞n=1 [tan-1 (n + 1) – tan-1 n]
> Find the sum of the series. ∑∞n=1 (-1)n πn/32n (2n)!
> Determine whether the series is convergent or divergent. ∑∞n=1 (-5)2n/n29n
> Determine whether the series is convergent or divergent. ∑∞n=1 1 ∙ 3 ∙5 ∙ ∙∙∙ (2n – 1)/5nn!
> Determine whether the series is convergent or divergent. ∑∞n=1 cos 3n/1 + (1.2)n
> Determine whether the series is convergent or divergent. ∑∞n=1 (-1)n-1 √n/n + 1
> Determine whether the series is convergent or divergent. ∑∞n=1 ln (n/3m + 1)
> Determine whether the series is convergent or divergent. ∑∞n=1 1/n √ln n
> Determine whether the series is convergent or divergent. ∑∞n=1 (-1)n/√n + 1
> Sketch the curve and find the area that it encloses. r2 = 4 cos 2θ
> Determine whether the series is convergent or divergent. ∑∞n=1 n3/5n
> Determine whether the series is convergent or divergent. ∑∞n=1 n2 +1/n3 + 1
> Determine whether the series is convergent or divergent. ∑∞n=1 n/n3 + 1
> A sequence is defined recursively by the equations a1 = 1, an+1 = 1/3(an + 4). Show that {an} is increasing and an < 2 for all n. Deduce that {an} is convergent and find its limit.
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. {(1 + 3/n)tr}
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. In n an
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. n sin n an ,2 + 1
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. an = = cos(nT/2)
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 3 n a 1+ п? ,2
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 94+1 an 10"
> Find the area of the shaded region. r= sin 20
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 2 + n3 1+ 2n3
> The force due to gravity on an object with mass m at a height above the surface of the earth is F = mgR2/ (R + h)2 where R is the radius of the earth and is the acceleration due to gravity. (a). Express F as a series in powers of h/R. (b). Observe that i
> This project deals with the function 1. Use your computer algebra system to evaluate f (x) for x = 1, 0.1, 0.01, 0.001 and 0.0001. Does it appear that f has a limit as x→0? 2. Use the CAS to graph f near x = 0. Does it appear that f h
> Write the binomial series expansion of (1 + x)k. What is the radius of convergence of this series?
> (a). Write an expression for the nth-degree Taylor polynomial of f centered at a. (b). Write an expression for the Taylor series of centered at f. (c). Write an expression for the Maclaurin series of f. (d). How do you show that f (x) is equal to the sum
> Suppose f (x) is the sum of a power series with radius of convergence R. (a). How do you differentiate f? What is the radius of convergence of the series for f'? (b). How do you integrate f? What is the radius of convergence of the series for ff (x) dx?
> (a). If a series is convergent by the Integral Test, how do you estimate its sum? (b). If a series is convergent by the Comparison Test, how do you estimate its sum? (c). If a series is convergent by the Alternating Series Test, how do you estimate its s
> (a). What is an absolutely convergent series? (b). What can you say about such a series?
> Suppose ∑an = 3 and sn is the nth partial sum of the series. What is limn lim n→∞ an? What is limn lim n→∞ sn?
> (a). What is a geometric series? Under what circumstances is it convergent? What is its sum? (b). What is a p-series? Under what circumstances is it convergent?
> Find the area of the shaded region. r= 4+3 sin e
> For the limit illustrate Definition 1 by finding values of that correspond to c = 1 and c = 0.1. lim (4 + x – 3x') = 2
> (a). What is a bounded sequence? (b). What is a monotonic sequence? (c). What can you say about a bounded monotonic sequence?
> (a). What is a convergent sequence? (b). What is a convergent series? (c). What does lima→∞ an = 3 mean? (d). What does ∑∞n=1 an = 3 mean?
> Write the Maclaurin series and the interval of convergence for each of the following functions. – x) (a) 1/(1 – x) (c) sin x (e) tan-'x (b) e* (d) cos x (f) In(1 + x)
> (a). Write the general form of a power series. (b). What is the radius of convergence of a power series? (c). What is the interval of convergence of a power series?
> State the following. (a). The Test for Divergence (b). The Integral Test (c). The Comparison Test (d). The Limit Comparison Test (e). The Alternating Series Test (f). The Ratio Test
> Any object emits radiation when heated. A blackbody is a system that absorbs all the radiation that falls on it. For instance, a matte black surface or a large cavity with a small hole in its wall (like a blast furnace) is a blackbody and emits blackbody
> (a). Show that the Maclaurin series of the function f (x) = x / 1 – x – x2 is ∑∞n=1 fn xn where fn is the nth Fibonacci number, that is, f1 = 1, f2 = 1, and fn = fn-1 + fn-2 for n > 2. [Hint: Write x/ (1 – x – x2) = c0 + c1x + c2x2 + … and multiply both
> Consider the series whose terms are the reciprocals of the positive integers that can be written in base 10 notation without using the digit 0. Show that this series is convergent and the sum is less than 90.
> Right-angled triangles are constructed as in the figure. Each triangle has height 1 and its base is the hypotenuse of the preceding triangle. Show that this sequence of triangles makes indefinitely many turns around P by showing that ∑
> Find all the solutions of the equation Hint: Consider the cases x > 0 and x 1 + 2! +... = 8! 4! + +
> Find the area of the shaded region. r=1+ cos 0
> Starting with the vertices P1 (0, 1), P2 (1, 1), P3 (1, 0), P4 (0, 0) of a square, we construct further points as shown in the figure: P5 is the midpoint of P1P2,P6 is the midpoint P2P3,P7 of is the midpoint of P3P4, and so on. The polygonal spiral path
> Find the sum of the series∑∞n=1 (-1)n/(2n + 1)3n.
> A sequence {an} is defined recursively by the equations ao = aj = 1 п(п — 1)а, — (п — 1)(п — 2)а,-1 — (п — 3)а,-2 %3D
> Suppose that circles of equal diameter are packed tightly in 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 rows of circles, show that An lim A
> If p > 1, evaluate the expression
> Let Show that u3 + v3 + w3 - 3uvw = 1. .3 u = 1 + 3! 6! 9! 10 v = x + 4! 7! 10! w = 2! 5! 8! + + + +
> 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 sum of the series ∑∞n=1 ln (1 – 1/n2).
> 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+ 3 4 6 8 12 +
