Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
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Problem 1. Let {r,} be a sequence and L be a real number. Give the definition that lim, In L. Prove from the definition of the limit, that 2n2 + 1 lim nx 4n? - n + 1 %3D by completing the following steps. (a) Using the fact that 1 <n < n?, estimate from above the expression 2n? +1 4n2 – n+1 b) Given e > 0 find a threshold N, so that for all n...
The work provided for part (b) was not correct.
(a) Suppose lim(Fm) = 1. Prove or disprove: There exists no E N such that IFml > 0.99 for all o (b) Prove or disprove:If (an) converges to a non-zero real number and (anbn) is convergent, then (bn) is convergent. RUP ) Let an→ L,CO) and an bn→12 n claim br) comvetgon Algebra of sesuenes an
(a) Suppose lim(Fm) = 1. Prove or disprove: There exists no E N such that...
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
Question 23.5
theorem
We have proved Theorem 1 for the special case T, but the exact same construction works for every R if we replace T with r in each step of the argument. However, if x e {+0, -00}, the argument requires some modification Question 23.5: Explain how you would modify the above argument to obtain a bijection f:NN such that f(n)-0 Theorem 1: Let nbe a series of real numbers that is only conditionally convergent, but not absolutely...
(2.2) Let a be a real number with 1<a< 2. Put f(x) = Q +r 1+2 (a) Show that f maps (1, 0) into (1, 0). (b) Show that f is a contraction on [1, ) and find its fixed point.
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% " (b) If CR is the semicircle of radius R in the upper half plane with centre at z = 0, show carefully that e2iz lim R00JCR (1+ z2)2 d% = 0 (c) Use residue calculus to evaluate the real integral I of part (a)
5. (a) Show that the following improper real integral is absolutely convergent cos 2x dr I (1+?}% "...
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above, x = lim sup (r) if and only if For all 0 there is an NEN, such that x <x+e whenevern > N, and b. For all >0 and all M, there is n > M with x - e< In a.
Example: Let {xn} be a sequence of real numbers. Show that Proposition 0.1 1. If r is bounded above,...
10. Read through the following "e-free" proof of the uniform convergence of power series. Does it depend on limn→oo lan|1/n or lim supn→oo lan! an)1/n? Explain. 1.3 Theorem. For a given power series Σ ak-a)" define the number R, 0 < R < oo, by n-0 lim sup |an| 1/n, then (a) if |z- a < R, the series converges absolutely (b) if lz-a > R, the terms of the series become unbounded and so the (c) if o<r <...