for these 2 theorems, pick on hypothesis, remove it, and provide a counterexample showing that the new statement is false.
for these 2 theorems, pick on hypothesis, remove it, and provide a counterexample showing that the new statement is false. (d) Thm The Weierstrass Uniform Convergence Criterion): The sequence...
for these 2 theorems, pick one hypothesis, remove it from the theorem to create a new statement. then, provide a counterexample showing that the new statement is false. (a) Thim (The Lagrange Remainder Theorem): Suppose f : 1 → R has n + 1 derivatives and ro e 1, Then for each r e I with r / ro there is some z strictly between ro and r with f(x) Pn (z) +Rn(x) where Rn is the remainder and ro)#1...
(1) Prove the Abel criterion for uniform convergence: 4. Suppose the series of functions 2 (x converges uniformly in some interval I. Suppose for every x E I, san(x)) forms a monotonic series, and that there is a constant K such that Then the series converges uniformly in I. (2) Using Abel criterion, compute the following limit: m- n 1 rn n= (1) Prove the Abel criterion for uniform convergence: 4. Suppose the series of functions 2 (x converges uniformly...
PLEASE use the THEORY below to give PROOF STEP BY STEP. This is an analysis class. Thank you. application of power series\Weierstrass M-test\term by term differentiability of power series sequence and series of function: pointwise and the theorem of uniform convergence which function is integrable: continuous and monotone Fri 19 Apr: The Fundamental Theorem of Calculus. (§7.5.) Wed 17 Apr: Example (∫10x2dx=1/3∫01x2dx=1/3). Basic properties of the integral. (mostly Theorem 7.4.2.) Fri 12 Apr: More on integrability, basic properties of the...