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mathematics
calculus

Questions and Answers of Calculus

Draw a picture to show thatWhat can you conclude about the series?
Explain why the Integral Test can't be used to determine whether the series is convergent.
Find the values of p for which the series is convergent.a.b.
Use the Integral Test to determine whether the series is convergent or divergent.a.b.c.
The Riemann zeta-function ( is defined byAnd is used in number theory to study the distribution of prime numbers. What is the domain of (?
Euler also found the sum of the p-series with p = 4:Use Euler's result to find the sum of the series. (a) (b)
(a) Use the sum of the first 10 terms to estimate the sum of the seriesHow good is this estimate? (b) Improve this estimate using 3 with n = 10. (c) Compare your estimate in part (b) with the exact
EstimateCorrect to five decimal places.
Show that if we want to approximate the sum of the seriesSo that the error is less than 5 in the ninth decimal place, then we need to add more than 1011,301 terms!
(a) Use 4 to show that if sn is the nth partial sum of the harmonic series, then Sn ≤ 1 + 1n n (b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first
Find all positive values of b for which the seriesconverges.
Determine whether the series is convergent or divergent.(a)(b) 1 + 1/8 + 1/27 + 1/64 + 1/125 + ...(c) 1 + 1/3 + 1/5 + 1/7 + 1/9 + ...
Suppose ∑ an and ∑ ba are series with positive terms and ∑ bn is known to be convergent. (a) If an > bn for all n, what can you say about ∑ an? Why? (b) If an < bn for all n, what can you say
Determine whether the series converges or diverges.(a)(b) (c)
Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.(a)(b)
The meaning of the decimal representation of a number 0.d1d2d3 (where the digit d1 is one of the numbers 0, 1, 2,...,9) is that 0.d1d2d3...= d1/10 d2/102 + d3/103 + d4/104 + ... Show that this
Prove that if an ≥ 0 and ∑ an converges, then also converges.
(a) Suppose that ˆ‘an and ˆ‘bn are series with positive terms and ˆ‘bn is divergent. Prove that ifThen ˆ‘an is also divergent.(b) Use part (a) to show that the series diverges.(i)(ii)
Show that if an > 0 and lim n→( nan ( 0, then ∑ an is divergent.
If ∑an is a convergent series with positive terms, is it true that ∑sin (an) is also convergent?
(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 terms?
Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy?a.(| error | b. (| error |
Approximate the sum of the series correct to four decimal places.a.b.
a.b. c.
Is the 50th partial sum S50 of the alternating series
For what values of is each series convergent?
Show that the series ( (- 1) n-1 bn, where bn = 1/n if n is odd and bn = 1/n2 if n is even, is divergent. Why does the Alternating Series Test no apply?
What can you say about the series (an in each of the following cases?(a)(b) (c)
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.a.b.c.
a. The terms of a series are defined recursively by the equationsa1 = 2Determine whether (an converge or diverges.
Let {bn} a sequence of positive numbers that converges to 1/2. Determine whether the given series is absolutely convergent.
For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer)?(a)(b) (c) (d)
(b) Deduce that lim n†’( xn/n! = 0 for all x.
(b) Find a value of n so that sn is within 0.00005 of the sum. Use this value of n to approximate the sum of the series.
Prove the Root Test Take any number such that L whenever n ( N.(i) If then use series is absolutely convergent (and therefore convergent)?(ii) If 1 or then the series is divergent?(iii) If the
Given any series (an, we define a series whose terms are all the positive terms of ( an and a series whose terms are all the negative terms of (an. to be specific, we letIf an > 0, then and
Suppose that series ( an is conditionally convergent.(a) Prove that the series (n2 an is divergent.(b) Conditional convergence of (an is not enough to determine whether (nan is convergent. Show this
Test the series for convergence or divergence.a.b. c.
(a)
Find the radius of convergence and interval of convergence of the series?a.b.c.
If k is a positive integer, find the radius of convergence of the series
Is it possible to find a power series whose interval of convergence is [0, (]? Explain.
The function J1 defined byIs called the Bessel function of order 1.(a) Find its domain.(b) Graph the first several partial sums on a common screen.(c) If your CAS has built-in Bessel functions, graph
A function f is defined byf(x) = 1 + 2x + x2 + 2x3 + x4 + ...that is, its coefficients are c2n = 1 and c2n+1 = 2 for all n ( 0. Find the interval of convergence of the series and find an explicit
Show that if where c ( 0, then the radius of convergence of the power series (cnXn is R = 1/c?
Suppose the series (cnXn has radius of convergence 2 and the series ( dnXn has radius of convergence 3. What is the radius of convergence of the series ((cn + dn)Xn?
If the radius of convergence of the power series is 10, what is the radius of convergence of the series ? way?
Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence.
(a) Use differ nation to find a power series representation forWhat is the radius of convergence? (b) Use part (a) to find a power series for (c) Use part (b) to find a power series for
Find a power series representation for the function the determine the radius of convergence.a. f(x) = 1n (5 - x)b.
Find a power series representation for f, and graph f and several partial sums sn(x) on the same screen. What happens as n increases?a.b.
Evaluate the indefinite integral as a power series. What is the radius of convergence?a.b.
Use a power series to approximate the definite integral to six decimal places.a.b.
Find a power series representation for the function and determine the interval of convergence.a.b. c.
Use the result of Example 7 to compute arctan 0.2 correct to five decimal places?
(a) Show that J0 (the Bessel function) of order 0 given in Example 4) satisfies the differential equation
(a) Show that the function
LetFind the intervals of convergence for f, f', and f".
Use the power series for tan-1 x to prove the following expression for ( as the sum of an infinite series:
Write the Maclaurin series and the interval of convergence for each of the following functions.(a) 1/ (1 - x)(b) ex(c) sin x(d) cos x(e) tan-1x(f) 1n(1 + x)
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?(g) The Root Test?
Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.a.b.c.
Determine whether the series is convergent or divergent.a.b. c.
Determine whether the series is conditionally convergent, absolutely convergent, or divergent.a.b.
Find the sum of the series.a.b. c.
Show that cosh x ( 1 + ½ x2 for all x.
Find the sum of the seriesCorrect to four decimal places.
Use the sum of the fist eight terms to approximate the sum of the series Estimate the error involved in this approximation?
Is also absolutely convergent?
Find the radius of convergence and interval of convergence of the series.a.b.
Find the Taylor Series of f(x) = sin x at a = (/6?
Find the Maclurni 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
Evaluate (ex / x dx as an infinite series?
(a) Approximate f by a Taylor polynomial with degree n 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)
Use series to evaluate the following limit?
(a) If f is an odd function, show thatC0 = c2 = c4 = ... = 0(b) If f is an even function, show thatC1 = c3 = c5 = ... = 0
A sequence is defined recursively by the equations a1 = 1, an + 1 = 1/3 (an + 4). Show that {an} is increasing and an < 2. Show that {an} is increasing and for all. Deduce that is convergent and find
If f (x) = sin (x3), find f (15) (0)
Find the sum of the series
If the curve y = e-x/10 sin x, x ( 0, is rotated about the x-axis, the resulting solid books like an infinite deceasing string of beads. (a) Find the exact volume of the nth bead. (Use either a table
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
Taking the value of xx at 0 to be 1 and integrating a series term by term, show that
Find the sum of the series (- 1)n / (2n + 1)3n
Find all the solutions of the equation Consider the cases x ( 0 and x < 0 separately.
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
LetShow that u3 + v3 + w3 - 3uvw = 1.
(a) Show that tan ½ x = cot ½ x - 2 cot x.(b) Find the sum of the series
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
(a) Show that for xy ( - 1.arctan x - arctan y = arctan x - y/1 + xyIf the left side lies between - (/2 and = (/4.(b) Show that arctan 120/119 - arctan 1/239 = (/4.(c) Deduce the following formula of
Find the interval of convergence of and find its sum?
Let A : = {k : k ∊ N; k < 20}; B : = {3k 1 : k ∊ N]; and C : = {2k + 1 : k ∊ N}: Determine the sets: (a) A ∩ B ∩ C, (b) (A ∩ B) \ C, (c) (A ∩ C) \ B.
Let f(x) := 1/x2; x ≠ 0; x ∈ R. (a) Determine the direct image f (E) where E := {x ∈ R : 1 < x < 2}. (b) Determine the inverse image f1(G) where G := {x ∈ R : 1 < x < 4}.
Let g(x) := x2 and f (x) := x + 2 for x ∈ R, and let h be the composite function h := g ο f. (a) Find the direct image h(E) of E := {x ∈ R : 0 < x < 1}. (b) Find the inverse image h1(G) of G :=
If f : A → B and E, F are subsets of A, then f (E ⋃ F) = f(E) ⋃ f(F) and f (E \ F) ⊂ f(E) ∩ f (F).
Show that if f : A → B and G, H are subsets of B, then f-1(G ⋃ H) = f-1 (G) ⋃ f-1(H) and f-1 (G ∩ H ) = f-1 (G) ∩ f-1(H).
Show that the function f defined by f (x) := x = x/√x2 + 1; x ∈ R, is a bijection of R onto {y : 1 < y < 1}.
(a) Give an example of two functions f, g on R to R such that f ≠ g, but such that f ο g = g ο f. (b) Give an example of three functions f, g, h on R such that f ο (g + h) ≠ f ο g) + f ο h.
(a) Show that if f : A → B is injective and E ⊂ A, then f-1( f (E)) = E. Give an example to show that equality need not hold if f is not injective. (b) Show that if f : A → B is surjective and
(a) Suppose that f is an injection. Show that f-1 ο f (x) = x for all x ∈ D(f) and that f ο f-1(y) = y for all y ∈ R( f ). (b) If f is a bijection of A onto B, show that f-1 is a bijection of B
Prove that if f : A → B is bijective and g : B → C is bijective, then the composite g ο f is a bijective map of A onto C.

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