If r = 0 then rn = 0 for all n ∈ IN. Obviously
This shows limn→∞ ax n = Lx. 19.3. Let 0 ≤ α < 1 and let f be a function from R → R which satisfies.
where R = 0 if the lim sup diverges to ∞ and R = ∞ if the lim sup is 0. Proof. Let r = lim sup n→∞.
The functions in Example 5.5 converge uniformly to 0 on R since.
Suppose that (fn) is a sequence of functions fn : A → R and f : A → R. Then fn → f uniformly on A if for every ϵ > 0
n→∞ xn+1 xn. = λ there exists n0 such that xn+1 xn. < r for all n ≥ n0. Hence
x = 0. 0 x = 0 for all x ∈ R. a. Show that h(n)(0) = 0 for all n ∈ N. b. Suppose x = 0. Show that the remainder term obtained by applying the Taylor's Theorem
converges for all x so R = ∞ for the power series ∑n≥0. (−1)n22n+1. (2n+1)! mainder Formula ((11) in Section 9.9) to show that limn→∞ rn(x) = 0 for all x ...
Does n0 0 uniformly on R?
n! 0 uniformly on R but that the sequence of derivatives (h0 n ) diverges for every x2R. Proof. Given >0; Choose N2N such that
Is n(x) dierentiable everywhere?
n(x) converges to f(x) = 0 uniformly, since taken >0;there exists N>2 , N2N such that for any nN;and for any x2R jf n(x) 0j 1 n jg(x)j 2 n 2 N f nis nowhere di erentiable for every n2N; but f(x) = 0 and hence is di erentiable everywhere. b)
Is H0 N(x) uniform?
If h0 n (x) ! g(x) is uniform in any interval containing 0 and also given each h0 n (x) is continuous at 0, implies g(x) should be continuous at x= 0;which is not the case. Hence the convergence is not uniform. Exercise 2 (6.3.4) Let h n(x) = sin(nx) p n Show that h n! 0 uniformly on R but that the sequence of derivatives (h0 n ) diverges for
Does show g(x) = limh0 n(x)?
Show g(x) = limh0 n (x) exists for all x, and explain how we can be certain that the convergence is not uniform on any neighborhood of zero. Proof. a) lim n!1 h
Chapter 2. Sequences §1. Limits of Sequences Let A be a nonempty