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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

We estimate integrals using Taylor polynomials. Exercise 72 is used to estimate the error. Let Q(x) = 1-x2²/6. Use the Error Bound for f(x) = sin x to show that Then calculate sin x X 5! --00x)/5 1x1/4 ·S (sin x/x) dx and find a bound for the error. Q(x) dx as an approximation to
Use the Squeeze Theorem to evaluate by verifying the given inequality. lim an n→∞
Determine convergence or divergence using any method covered so far. 8 n=1 1 n³/2 - (Inn)4
Use the result of Exercise 73 and your knowledge of alternating series to show thatData From Exercise 73When a voltage V is applied to a series circuit consisting of a resistor R and an inductor L, the current at time t isExpand I(t) in a Maclaurin series. Show that I(t) ≈ Vt/L for small t.
Apply the Root Test to determine convergence or divergence, or state that the Root Test is inconclusive. 00 n=1 n n
Determine convergence or divergence using any method covered so far. Σ n=2 4η2 + 15η 3n4 – 5n2 – 17
Which of the following statements is equivalent to the assertion Explain. lim an = L? n→∞
Apply the Root Test to determine convergence or divergence, or state that the Root Test is inconclusive. 00 n=1 3 An n
Find the Maclaurin series for ƒ(x) = cos(x3) and use it to determine ƒ(6)(0).
Determine convergence or divergence using any method covered so far. 00 8 Σ n=1 n 4-n + 5-n
Prove by induction that for all k,Use this to prove that Tn agrees with ƒ at x = a to order n. di ((x-a)k) dxj k! = di ((x-a)k) dxj k(k-1) (kj+ 1)(x-a)k-j k! a* L = 6 k! 1 for k=j 0 for kj
Show that an = 1/2n + 1 is decreasing.
Find ƒ(7)(0) and ƒ(8)(0) for ƒ(x) = tan−1 x using the Maclaurin series.
Show that an = 3n2/n2 + 2 is increasing. Find an upper bound.
Let a be any number and let P(x) = an xn + an−1xn−1 + · · · + a1x + a0 be a polynomial of degree n or less.(a) Show that if P(j)(a) = 0 for j = 0, 1, . . . , n, then P(x) = 0, that is, aj = 0 for all j. Use induction, noting that if the statement is true for degree n − 1, then P'(x) =
Determine convergence or divergence using any method covered in the text. 00 Σ n=1 1 n √n + Inn
Let cn = 1 + 1 + 1 + 1 + 2 + · · · + 1/2n. (a) Calculate C₁, C2, C3, C4. (b) Use a comparison of rectangles with the area under y = x-¹ over the interval [n, 2n] to prove that 1 is the dx + !! 82" dx + 21/11 ≤ Cn ≤ X 2n X n n (c) Use the Squeeze Theorem to determine lim Cn. n→∞0
Determine convergence or divergence using any method covered in the text. Μ8 n=1 1 √n | 1 Vn + 1
Let an = Hn − ln n, where Hn is the nth harmonic number: H₂ = 1 + 2 + + 3 n+1 Sa+s (a) Show that an ≥ 0 for n ≥ 1. Hint: Show that Hn > dx X n
Determine convergence or divergence using any method covered in the text. M8 cos(πη) Σ n2/3 n=2
Determine convergence or divergence using any method covered in the text. 00 n=1 (2) alm 3 n
Use substitution to find the first three terms of the Maclaurin series for ƒ(x) = ex20. How does the result show that ƒ(k)(0) = 0 for 1 ≤ k ≤ 19?
For which a doesconverge? 00 h=2 1 na Inn
Show that an = 3√n + 1 − n is decreasing.
Determine convergence or divergence using any method covered in the text. 00 Σ n=1 7η π e8n
Use the binomial series to find ƒ(8)(0) for ƒ(x) = √1 − x2.
For which values of p doesconverge? 00 ne Σ (n3 + 1)P n=1
Determine convergence or divergence using any method covered in the text. 00 Σε n=1 W -0.02n
Does the Maclaurin series for ƒ(x) = (1 + x)3/4 converge to ƒ(x) at x = 2? Give numerical evidence to support your answer.
For which values of p doesconverge? 00 et (1 + e²x)p n=1
Give an example of divergent sequences {an} and {bn} such that {an + bn} converges.
Determine convergence or divergence using any method covered in the text. 00 Ene n=1 ,-0.02n
Explain the steps required to verify that the Maclaurin series for ƒ(x) = ex converges to ƒ(x) for all x.
Determine convergence or divergence using any method covered in the text. Μ8 (−1)"-1 γh + Vn + 1 Σ n=1 V
Let an = ƒ (n), where ƒ is a continuous, decreasing function such that ƒ (x) ≥ 0 andShow that S₁. f(x) dx.
Determine convergence or divergence using any method covered in the text. 00 Σ n=10 1 n(Inn)3/2
Let ƒ(x) = √1 + x.(a) Use a graphing calculator to compare the graph of ƒ with the graphs of the first five Taylor polynomials for ƒ. What do they suggest about the interval of convergence of the Taylor series?(b) Investigate numerically whether or not the Taylor expansion for ƒ is valid for
Use the first five terms of the Maclaurin series for the elliptic integral E(k) to estimate the period T of a 1-m pendulum released at an angle θ = π/4 (see Example 12). EXAMPLE 12 Elliptic Function Find the Maclaurin series for E(k) and estimate E(k) for k = sin
Let an = ƒ (n), where ƒ is a continuous, decreasing function such that ƒ (x) ≥ 0 andUsing the inequality in (3), show thatThis series converges slowly. Use a computer algebra system to verify that SN 43,129 ≈ 5.00000021. S₁. f(x) dx.
Use Example 12 and the approximation sin x ≈ x to show that the period T of a pendulum released at an angle θ has the following second-order approximation: EXAMPLE 12 Elliptic Function Find the Maclaurin series for E(k) and estimate E(k) for k= sin.
Let an = ƒ (n), where ƒ is a continuous, decreasing function such that ƒ (x) ≥ 0 andData From Exercise 81Let an = ƒ (n), where ƒ is a continuous, decreasing function such that ƒ (x) ≥ 0 andShow that S₁. f(x) dx.
Determine convergence or divergence using any method covered in the text. 00 (-1)" Σ Inn n=2
Theorem 1 states that if then the sequence an = ƒ(n) converges and Show that the converse is false. In other words, find a function ƒ such that an = ƒ(n) converges but does not exist. lim f(x) = L, X-80
Use the limit definition to prove that if {an} is a convergent sequence of integers with limit L, then there exists a number M such that an = L for all n ≥ M.
Determine convergence or divergence using any method covered in the text. 00 n! Σ (2η)! n=1
The limits can be done using multiple L’Hôpital’s Rule steps. Power series provide an alternative approach. In each case substitute in the Maclaurin series for the trig function or the inverse trig function involved, simplify, and compute the limit. cosx−1+ 14 lim X-0 2
Use the inequalities in (4) from Exercise 83 with M = 43,129 to prove thatData From Exercise 83 (4) M Σan n=1 JM+1(x) dx ≤S ≤ + M+1 Σan + n=1 So f(x) dx M+1
Determine convergence or divergence using any method covered in the text. Σ n=2 n 1 + 100m
Use the limit definition to prove that the limit does not change if a finite number of terms are added or removed from a convergent sequence.
The limits can be done using multiple L’Hôpital’s Rule steps. Power series provide an alternative approach. In each case substitute in the Maclaurin series for the trig function or the inverse trig function involved, simplify, and compute the limit. lim x→0 sin x - x + x5 6
Use the inequalities in (4) from Exercise 83 with M = 40,000 to show thatIs this consistent with Euler’s result, according to which this infinite series has sum π2/6?Data From Exercise 83 (4) 1.644934066 ≤ n=1 n² 1.644934068
Determine convergence or divergence using any method covered in the text. Σ n=2 n3 – 2n2 + n - 4 2n4 + 3η3 – 4n2 – 1
The limits can be done using multiple L’Hôpital’s Rule steps. Power series provide an alternative approach. In each case substitute in the Maclaurin series for the trig function or the inverse trig function involved, simplify, and compute the limit. lim X-0 tan-¹ x - x cos x x5 - 1x³
Use a CAS and the inequalities in (5) from Exercise 83 to determine the value of within an error less than 10−4. Check that your result is consistent with that of Euler, who proved that the sum is equal to π6/945.Data From Exercise 83 (5) 18 Ση η=1 -6 to
Let {an} be a sequence such that exists and is nonzero. Show that exists if and only if there exists an integer M such that the sign of an does not change for n > M. lim anl n→∞o
Determine convergence or divergence using any method covered in the text. 00 COS n Σ η3/2 n=1
Use a CAS and the inequalities in (5) from Exercise 83 to determine the value of within an error less than 10−4.Data From Exercise 83 (5) 00 της H=1 -5 to
The limits can be done using multiple L’Hôpital’s Rule steps. Power series provide an alternative approach. In each case substitute in the Maclaurin series for the trig function or the inverse trig function involved, simplify, and compute the limit. lim x-0 sin(x²) x4 cos x) x²
How far can a stack of identical books (of mass m and unit length) extend without tipping over? The stack will not tip over if the (n + 1)st book is placed at the bottom of the stack with its right edge located at or before the center of mass of the first n books (Figure 6). Let cn be the center of
Proceed as in Example 13 to show that the sequence is increasing and bounded above by M = 3. Then prove that the limit exists and find its value.Example 13 √3, 3 V3, V3 √3 V3, ...
Let {an} be the sequence defined recursively by ao = 0, an+1 = √2+ an
Determine convergence or divergence using any method covered in the text. 00 Σ n=1 n Vm3/2 + 1
The following argument proves the divergence of the harmonic serieswithout using the Integral Test. To begin, assume that the harmonic series converges to a value S. 00 η=1 1/n
Show that Verify that n! ≥ (n/2)n/2 by observing that half of the factors of n! are greater than or equal to n/2. lim Vn! = ∞o. 00. n→∞0
Use Euler’s Formula to express each of the following in a + bi form.(a) eπ/4i (b) 4e5π/3i (c) ie−π/2 i
Determine convergence or divergence using any method covered in the text. Σ(52) n=1 M
Consider the series an, where an = (ln(ln n))− ln n. 00 Σ n=2 an
Use Euler’s Formula to express each of the following in a + bi form.(a) −e3π/4 i (b) e2πi (c) 3ie−π/3 i
Determine convergence or divergence using any method covered in the text. 00 n=1 en n!
Suppose we wish to approximate There is a similar telescoping series whose value can be computed exactly (Example 2 in Section 10.2):Example 2 Section 10.2 00 S s = Σ 1/n?. n=1
Let bn = n√n!. (a) Show that In bn = n k=1 k In -. n ·S' (b) Show that In bn converges to bn → e-¹. In xdx, and conclude that
Use Euler’s Formula to prove that the identity holds. Similarity between these relationships and the definitions of the hyperbolic sine and cosine functions. COS Z = eiz te-iz 2
Given positive numbers a1 (a) Show that an ≤ bn for all n (Figure 14).(b) Show that {an} is increasing and {bn} is decreasing.(c) Show that bn+1 − an+1 ≤ bn − an/2.(d) Prove that both {an} and {bn} converge and have the same limit. This limit, denoted AGM(a1, b1), is called the
The sum has been computed to more than 100 million digits. The first 30 digits are S = 1.202056903159594285399738161511. Approximate S using Kummer’s Acceleration Method of Exercise 91 with the similar series and M = 500. According to Exercise 60 in Section 10.2, the similar series is a
Use Euler’s Formula to prove that the identity holds. Similarity between these relationships and the definitions of the hyperbolic sine and cosine functions. sin z = eiz-e-iz 2i
Determine convergence or divergence using any method covered in the text. 00 Σ n=1 1 {n(1 + Vn) }\ $00 n=1 1 {n(1 + Vn) }\$
In this exercise, we show that the Maclaurin expansion of ƒ(x) = ln(1 + x) is valid for x = 1. (a) Show that for all x # -1, (b) Integrate from 0 to 1 to obtain 1 1 + x In 2 = = N Σ n=1 N Σ(-1)"x" + n=0 (-1)n-1 n In 2 = 1 - + (-1) N+1 1 (c) Verify that the integral on the right tends to zero as
Let g(t) = 1/1 + t2 − t/1 + t2. S' (b) Show that g(t) = 1-t-t² +1³ + 14 - 15 - 16 + (c) Evaluate S = 1 -/- ++ (a) Show that g(t)dt = π - 4 2 In 2. - 5 6 7 +
Determine convergence or divergence using any method covered in the text. 00 Σ(Inn - In(n + 1)) n=1
We investigate the convergence of the binomial seriesProve that Ta(x) has radius of convergence R = 1 if a is not a whole number. What is the radius of convergence if a is a whole number? Ta(x) = M8 Σ n=0
We investigate the convergence of the binomial seriesBy Exercise 94, Ta(x) converges for |x| a(x) = (1 + x)a.Data From Exercise 94We investigate the convergence of the binomial seriesProve that Ta(x) has radius of convergence R = 1 if a is not a whole number. What is the radius of convergence if a
Determine convergence or divergence using any method covered in the text. 00 Σ n=1 1 n + \n Vn
The function is called an elliptic integral of the second kind. Prove that for |k| G(k)= π/2 0 √1-k² sin² tdt
Assume that a 2 + (y/b)2 = 1 shown in Figure 4. There is no explicit formula for L, but it is known that L = 4bG(k), with G(k) as in Exercise 96 and k = √1 − a2/b2. Use the first three terms of the expansion of Exercise 96 to estimate L when a = 4 and b = 5.Data From Exercise 96The function is
Determine convergence or divergence using any method covered in the text. 00 Σ n=2 1 nInn
Use Exercise 96 to prove that if a Use the first two terms of the series for G(k).Data From Exercise 96The function is called an elliptic integral of the second kind. Prove that for |k| π L = (36 2 (3b + b
Determine convergence or divergence using any method covered in the text. 00 1 Σ in n n=2
Determine convergence or divergence using any method covered in the text. 00 Σsin² n=1 π n
Prove that e is an irrational number using the following argument by contradiction.Suppose that e = M/N, where M, N are nonzero integers.(a) Show that M! e−1 is a whole number.(b) Use the power series for ƒ(x) = ex at x = −1 to show that there is an integer B such that M! e−1 equals(c)
Determine convergence or divergence using any method covered in the text. M8 n=0 22n n!
Use the result of Exercise 75 in Section 4.5 to show that the Maclaurin series of the functionData From Section 4.5 Exercise 75Sketch the graph of ƒ(x) = x2/x +1 Proceed as in the previous exercise to find the slant asymptote.is T(x) = 0. This provides an example of a function ƒ whose Maclaurin
Find the interval of convergence of the power series. 8 Σ n=0 20 χ n!
Find the interval of convergence of the power series. 00 Σ ux | n + 1 + n=0
Find the interval of convergence of the power series. Σ n=0 n n8 + 1 - (x − 3)” -
Find the interval of convergence of the power series. 00 Σηχ" nx n=0
Find the interval of convergence of the power series. 00 n=0 (nx)"
Find the interval of convergence of the power series. ∞ n=2 (2x - 3) nln n
Prove that Express the left-hand side as the derivative of a geometric series. n=0 -ηχ ne = e-x (1 – e-x)2
Expand ƒ(x) = 2/4 − 3x as a power series centered at c = 0. Determine the values of x for which the series converges.
Let (a) Show that F(x) has infinite radius of convergence.(b) Show that y = F(x) is a solution of(c) Plot the partial sums SN for N = 1, 3, 5, 7 on the same set of axes. F(x) = 00 Σ k=0 x24 2k 2k.k!

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