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An Introduction to Analysis 4th edition William R. Wade - Solutions

Suppose that f : [0, 1] †’ [a, b] is integrable on [0, 1]. Assume that ef(x) and |f(x)|p are integrable for all 0 a) Prove thatfor all 0 b) If 0 c) State and prove analogues of these results for improper integrals.
Let f be continuous on a closed, bounded interval [a, b] and suppose that DRf{x) exists for all x ∈ (a, b). a) Show that if f(b) < y0 < f(a), then x0 := sup{x ∈ [a, b] : f(x) > y0} satisfies f(x0) = y0 and DRf(x0) < 0. b) Prove that if f(b) < f(a), then there are uncountably many points x which
Let {ak} and be real sequences. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones.a) If ak is strictly decreasing and ak †’ 0 as k †’ ˆž, then ˆ‘ˆžk=1 converges.b) If ak ‰ bk for all k ˆˆ N
Prove that each of the following series converges and find its value,a)b)c)d)
A series ˆ‘ˆžk=0 ak is said to be Cesdro summable to an L ˆˆ R if and only ifconverges to L as n †’ ˆž.a) Let sn = ˆ‘n-1k=0 ak. Prove thatfor each n ˆˆ N.b) Prove that if ak ˆˆ R and ˆ‘ˆžk=0 ak = L converges, then ˆ‘ˆžk=0 ak is Cesaro summable to L.c) Prove that
Suppose that ak > 0 for k large and that ˆ‘ˆžk=1 ak/k converges. Prove that
If ˆ‘nk=1 = kak = (n + 1)/(n + 2) for ˆˆ N, prove that
Represent each of the following series as a telescopic series and find its value.a)b)c)d)
Prove that each of the following series diverges.a)b) c)
Let a0, a1,... be a sequence of real numbers. If ak †’ L as k †’ ˆž, doesConverge? If so, which is its value?
Find all x ˆˆ R for whichConverge? For each such x, find the value of this series.
a) Prove that if ∑∞k=1 ak converges, then its partial sums sn are bounded. b) Show that the converse of part a) is false. Namely, show that a series ∑∞k=1 ak may have bounded partial sums and still diverge.
Suppose that I is a closed interval and x0 ˆˆ I. Suppose further that f is differentiable on R, that f'(a) ‰ 0 for some a ˆˆ R, that the functionsatisfies F(I) Š‚, and that there is a number 0 a) Prove that |F(x) - F(y)| b) If xn:= F(xn-1) for n ˆˆ
a) Suppose that {ak} is a decreasing sequence of real numbers. Prove that if ∑∞k=1 ak converges, then kak → 0 as k → ∞. b) Let sn = ∑nk=1 (- l)k+l/k for n ∈ N. Prove that S2n is strictly increasing, S2n+1 is strictly decreasing, and s2n+1 - s2n → 0 as n → ∞. c) Prove that part
Let {bk} be a real sequence and b ˆˆ R.a) Suppose that there are M, N ˆˆ N such that |b - bk| N. Prove thatfor all n > N. b) Prove that if bk †’ b as k †’ ˆž, then as n †’ ˆž. c) Show that the converse of b) is false.
Let {ak} and {bk} be real sequences. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones. a) If ∑∞k=1 ak converges and ak/bk → 0 as k → ∞, then ∑∞k=1 bk converges. b) Suppose that 0 < a < l. If ak > 0
Prove that each of the following series converges.a)b) c) d) e) f)
Find all p ˆˆ R such thatconverges.
Prove that each of the following series diverges.a)b) c) d)
If ak > 0 is a bounded sequence, prove thatconverges for all p > 1.
Find all p > 0 such that the following series converges:
If ˆ‘ˆžk=1 |ak| converges, prove thatconverges for all p > 0. What happens if p
Suppose that ak and bk are nonnegative for all k ∈ N. Prove that if ∑∞k=1 ak and ∑∞k=1 bk converge, then ∑∞k=1 akbk also converges.
Suppose that a,b ˆˆ R satisfy b/a ˆˆ RZ. Find all q > 0 such thatconverges.
Suppose that ak → 0. Prove that ∑∞k=1 ak converges if and only if the series ∑∞k=1 (a2k + a2k+1) converges.
Let {ak} and be real sequences. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones.a) Suppose that 0 b) If ˆ‘ˆžk=1 ak is absolutely convergent and ak †“ 0 as k †’ ˆž,
Prove that each of the following series converges.a)b) c) d)
Let x
a) Using Exercise 4.4.4, prove thatfor all x ˆˆ [0, Ï€/2]. b) For x ˆˆ [0, Ï€/2]
Decide, using results covered so far in this chapter, which of the following series converge and which diverge.a)b) c) d) e) f) g)
For each of the following, find all values of p ˆˆ R for which the given series converges absolutely.a)b) c) d) e) f)
Suppose that ak > 0 and that ak1/k → as k → ∞. Prove that ∑∞k=1 akxk converges absolutely for all |x| < 1/a if a ≠ 0 and for all x ∈ R if a = 0.
Defines ak recursively by a1 = 1 andProve that ˆ‘ˆžk=1 ak converges absolutely.
Suppose that akj > 0 for k, j ˆˆ N. Setfor each k ˆˆ N, and suppose that ˆ‘ˆžk=1 Ak converges. a) Prove that b) Show that c) Prove that b) may not hold if akj has both positive and negative values. Consider
a) Suppose that ∑∞k=1 ak converges absolutely. Prove that ∑∞k=1 |ak|p converges for all p > 1. b) Suppose that ∑∞k=1 ak converges conditionally. Prove that ∑∞k=1 kpak diverges for all p > 1.
For any real sequence definea) Prove that if lim infk†’ˆžxk > x for some x ˆˆ R, then xk > x for k large. b) Prove that if xk †’ x as k †’ ˆž, for some x ˆˆ R, then lim infk†’ˆž xk = x. c) If ak > 0 for all k
Given that ˆ‘ˆžk=1 1/k2 = Ï€2/6 (see Exercise 14.3.7), find the exact value of
Let {ak) and [bk) be real sequences. Decide which of the following statements are true and which are false. Prove the true ones and give counterexamples to the false ones. a) If ak ↓ 0, as k → ∞, and ∑∞k=1 bk converges conditionally, then ∑∞k=1 akbk converges. b) If ak → 0, as k →
Prove that each of the following series converges.a)b) c) (d) e)
For each of the following, find all values x ˆˆ R for which the given series converges.a)b)c)d)
Using any test covered in this chapter so far, find out which of the following series converge absolutely, which converge conditionally, and which diverge.a)b) c) d) e)
[ABEL'S TEST] Suppose that ∑∞k=1 ak converges and that bk ↓ b as k → ∞. Prove that ∑∞k=1 akbk converges.
Show that under the hypotheses of Dirichlet's Test,
Suppose that {ak} and {bk} are real sequences such that ak †’ 0 as k †’ ˆž,Prove that ˆ‘ˆžk=1 akbk converges.
Suppose that ˆ‘ˆžk=1 ak converges. Prove that if bk †‘ ˆž and ˆ‘ˆžk=1 akbk converges, thenas m †’ ˆž.
Prove thatconverges for every x ˆˆ (0, 2Ï€) and every ak †“ 0. What happens when x = 0?
Suppose that ak †“ 0 as k †’ ˆž. Prove that
For each of the following series, let sn represent its partial sums and s its value. Prove that s is finite and find an n so large that sn approximates s to an accuracy of 10-2.a)b) c)
a) Find all p > 0 such that the following series converges:b) For each such p, prove that the partial sums of this series sn and its value s satisfy
For each of the following series, let sn represent its partial sums, and let s represent its value. Prove that s is finite and find an n so large that sn approximates s to three decimal places.a)b) c) d)
Prove Theorem 6.40ii.
Using any test covered in this chapter, find out which of the following series converge absolutely, which converge conditionally, and which diverge.a)b) c) d)
For each of the following, find all values of p ˆˆ R for which the given series converges absolutely, for which it converges conditionally, and for which it diverges.a)b) c)
a) Prove that the Root Test applied to the seriesyields r = 1. Use the Logarithmic Test to prove that this series converges. b) Prove that the Ratio Test applied to the series yields r = 1. Use Raabe's Test to prove that this series converges.
Suppose that f: R †’ (0, ˆž) is differentiable, that f(x) †’ 0 as x †’ ˆž, and thatexists. If a
Suppose that [ak] is a sequence of nonzero real numbers and thatexists as an extended real number. Prove that ˆ‘ˆžk=1 ak converges absolutely when p > 1.
a) Prove that x/n → 0 uniformly, as n → ∞, on any closed interval [a, b] b) Prove that 1/(nx) → 0 pointwise but not uniformly on (0, 1) as n → ∞.
Let E be a nonempty subset of R and f be a real-valued function defined on E. Suppose that fn is a sequence of bounded functions on E which converges to f uniformly on E. Prove thatuniformly on E as n †’ ˆž (compare with Exercise 6.1.9).
Let fn be integrable on [0, 1] and fn †’ f uniformly on [0, 1]. Show that if bn f †‘ 1 as n †’ ˆž, then
Prove that the following limits exist and evaluate them.a)
A sequence of functions fn is said to be uniformly bounded on a set E if and only if there is an M > 0 such that |fn (x) < M for all x ∈ E and all n ∈ N. Suppose that for each n ∈ N, fn: E → R is bounded. If fn → f uniformly on E, as n → N, prove that {fn} is uniformly bounded on E and
Let [a, b] be a closed bounded interval, f : [a, b] → R be bounded, and g : [a, b] → R be continuous with g(a) = g(b) = 0. Let fn be a uniformly bounded sequence of functions on [a, b] (see Exercise 7.1.3). Prove that if fn → f uniformly on all closed intervals [c, d] ⊂ (a, b), then fng →
Suppose that fn → f and gn → g, as n → ∞, uniformly on some set E ⊂ R. a) Prove that fn + gn → f + g and αfn → αf → αf, as n → ∞ uniformly on E for all α ∈ R. b) Prove that fngn → fg pointwise on E. c) Prove that if f and g are bounded on E, then fngn on E. then fngn →
Suppose that E is a nonempty subset of R and that fn → f uniformly on E. Prove that if each fn is uniformly continuous on E, then f is uniformly continuous on E.
Suppose that f is uniformly continuous on R. If yn → 0 as n → ∞ and fn(x) := f(x + yn) for x ∈ R, prove that fn converges uniformly on R.
Suppose that b > a > 0. Prove that a > 0. Prove that">
Let f, g be continuous on a closed bounded interval [a, b] with |g(x)| > 0 for x ∈ [a, b]. Suppose that fn → f and gn → g as n → ∞, uniformly on [a, b]. a) Prove that 1/gn is defined for large n and fn/gn → f/g uniformly on [a, b] as n → ∞. b) Show that a) is false if [a, b] is
a) Prove that ∑∞k=1 sin(x/k2) converges uniformly on any bounded interval in R. b) Prove that ∑∞k=0e-kx converges uniformly on any closed subinterval of (0, ∞).
Suppose that f1, f2,... are continuous real functions defined on a closed, bounded interval [a, b]. If 0
Prove that the geometric seriesconverges uniformly on any closed interval [a, b] Š‚ (-1, 1).
Let E(x) = ˆ‘ˆžk=0 xk/k!.a) Prove that the series defining E(x) converges uniformly on any closed interval [a, b].b) Prove thatfor all a, b ˆˆ R. c) Prove that the function y = E(x) satisfies the initial value problem y'-y = 0, y(0 ) = l. [We shall see in Section 7.4
Suppose thatProve that
Show thatconverges, pointwise on R and uniformly on each bounded interval in R, to a differentiable function f which satisfies |f(x) for all x ˆˆ R.
Prove that
Suppose that f = ∑∞k=1 fk converges uniformly on a set E ⊂ R. If g1 is bounded on E and gk(x) > gk+1(x) > 0 for all x ∈ E and k ∈ N, prove that ∑∞k=1 fkgk converges uniformly on E.
Let n > 0 be a fixed nonnegative integer and recall that 0! := 1. The Bessel function of order n is the function defined bya) Show that Bn(x) converges pointwise on R and uniformly on any closed interval [a,b. b) Prove that y = Bn(x) satisfies the differential equation x2y" + xy' + (x2 - n2)y =
Suppose that ak ↓ I 0 as k → ∞. Prove that ∑∞k=1 ak sin kx converges uniformly on any closed interval [a, b] ⊂ (0, 2π).
Find the interval of convergence of each of the following power series.a)b) c) d)
Suppose that ak †“ 0 as k †’ ˆž. Prove that given Îµ > 0 there is a Î´ > 0 such thatfor all x, y ˆˆ [0, 1] which satisfy |x - y|
a) Prove the following weak form of Stirling's Formula (compare with Theorem 12.73):b) Find all x ˆˆ R for which the power series converges absolutely.
Find the interval of convergence of each of the following power series.a)b) c) d)
Suppose that ∑∞k=0 akxk has radius of convergence R ∈ (0, ∞). a) Find the radius of convergence of ∑∞k=0 akx2k. b) Find the radius of convergence of ∑∞k=0 a2kxk .
Suppose that |ak| |bk| for large k. Prove that if ∑∞k=0 bkxk converges on an open interval I, then ∑∞k=0 akxk also converges on I. Is this result true if open is omitted?
A series ˆ‘ˆžk=0 ak is said to be Abel summable to L if and only ifa) Prove that if ˆ‘ˆžk=0 ak converges to L, then ˆ‘ˆžk=0 ak is Abel summable to L. b) Find the Abel sum of ˆ‘ˆžk=0 (-1)k.
Find a closed form for each of the following series and the largest set on which this formula is valid.a)b) c) d)
If ˆ‘ˆžk=1 akxk has radius of convergence R and ak ‰ 0 for large prove that
Prove thatis differentiable on (-3, 3) and for 0
Prove that each of the following functions is analytic on R and find its Maclaurin expansion. a) x2 + cos(2x) b) x23x c) cos2 x - sin2 x d) ex - 1/x
Suppose that f is analytic on (-ˆž, ˆž) and thatfor some a ‰ b in R. Prove that f(x) = 0 for all x ˆˆ R.
Prove thatfor all ak ˆˆ R and all > 1.
Prove that each of the following functions is analytic on (-1, 1) and find its Maclaurin expansion. a) x/x5 + 1 b) ex/1 + x c) log (1/|x2 - 1|) d) arcsin x
For each of the following functions, find its Taylor expansion centered at x0 = 1 and determine the largest interval on which it converges. a) ex b) log2(x5) c) x3 - x + 5 d) √x
Let a > 0 and suppose that f ∈ C∞(-a, a). a) If f is odd [i.e., if (-x) = -f(x) for all x ∈ (-a, a)], then the Maclaurin series of f contains only odd powers of x. b) If f is even [i.e., if f(-x) = f(x) for all x ∈ (-a, a)], then the Maclaurin series of f contains only even powers of x.
Suppose that f ˆˆ Cˆž(-ˆž, ˆž) and thatfor all a ˆˆ R. Prove that f is analytic on (-ˆž, ˆž) and
a) Prove thatfor n ˆˆ N. b) Show that
Let f ∈ C∞(a, b). Prove that f is analytic on (a, b) if and only if f' is analytic on (a, b).
Suppose that I is a nonempty open interval and that f is bounded and Cˆž on I. If there is an M > 0 such that |f(k)(x)|for n = 0, 1, 2, ..., then prove that f is zero on [a, b].
Using a calculator and Theorem 7.58, approximate all real roots of f(x) = x3 + 3x2 + 4x + 1 to five decimal places.
Using the proof of Theorem 7.58, prove that (20) holds if r/2 replaces r. Use part a) to estimate the difference |π4 - π|, where x0 = 3, f(x) = sin x, and xn is defined by (19). Evaluate x4 directly, and verify that x4 is actually closer than the theory predicts.
Prove that given any n ∈ N, there is a function f ∈ Cn(R) such that f(n+1)(x) does not exist for any x ∈ R.

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