f ** Let f E C[0, 1] be given by f(x) = e". For each n 2 0, let fn E C0, 1] be given by 7L fn(x) = T=0 (so fn(x) is the first n+1 terms in the Maclaurin Series expansion of e"). Show that the sequence (fn) converges to f. (Use the Lo metric.)

Q: Do the following sequences {a} converge or diverge? Justify your answers (a) an 3" (b) an - (-1)"/ñ…

A: Explanation of the answer is as follows

Q: (f) Assume that g(x) > bna" is convergent on (-R, R). Instead of assuming that f(rn) = 0 assume that…

A: We will prove the part (f) using two theorems   Theorem (a)Let f(x)=∑anxn be a non-constant power…

Q: p for which In (n) Comvenges

A: Given∑n=1∞ln(n)np

Q: Let fn RR be a sequence of continuous functions which converg uniformly to a function f : R → R. Let…

A: see my solution

Q: 5n 9. Use the L'Hopital's rule to determine if the sequence an = () converges or diverges as limit n…

A:

Q: 00 (-1)"+1 Is En=1 convergent absolutely, non-absolutely, or diverges. n+8

A: Given series is ∑n=1∞-1n+1n+8 Absolute convergent test: If ∑an converges, then ∑an converges. Let…

Q: Let the function g be defined by g(x) = (x^6 + 2)/7. Use the Fixed-Point Theorem to show that the…

A: This question is basically based on the theorem named Banach fixed point Theorem which states that:…

Q: Show that f"(0) = 0. Use induction and the ideas from the above to show that f(k)(0) = 0 for every k…

A: Given that fx=∑n=0∞anxn in the interval -R,R. Then R is the radius of convergence of the function f.…

Q: 2 ทะ n (8) n! h

A:

Q: 1.) Prove that the sequence An= nt n In (2+1) converges to O. That is, prove limln (^+1)=0 nyo

A:

Q: Show that >n"x" diverges for all x + 0. n=0

A:

Q: Is {3n/(3n-1)} n=1 to infinity, a convergent sequence in R with the usual metric dR(x,y) = abs(x-y)?

A: The given sequence is .We need to check that the given sequence is convergent or divergent with the…

Q: Suppose that an<0 for all n, and define b,=tan-(a,-n). Then the sequence {b,}... O diverges to - O…

A: Solve the following

Q: Solve Σ 0 to ∞ ((n! • x^n)/ n^n) for x = e and x = -e and explain whether each point converges or…

A: Solve for x = e and check whether it converges or diverges as follows.

Q: Theorem 72. (Products of two convergent sequences) Let {px} be a sequence that converges to x and…

A:

Q: 00 Let r be a constant. Suppose ' converges to -1, then the sum of )r' is i-1 i-0

A:

Q: Let Xn=nn for n EN * Show that Xn+₁ <Xn if and only if (1+1)^<N. *Show that the inequality is valid…

A:

Q: 24.6 Let fn(x) = (x − 1)² for x = [0, 1]. (a) Does the sequence (fn) converge pointwise on the set…

A:

Q: Study uniform convergence on 10, 1) of the sequence of functions Jn for the following: a fn(x) =…

A: Solution a: Given that fn(x)=11+(nx+1)3 in 0,1. Now, f(x)=limn→∞fn(x)=limn→∞11+(nx+1)3=0,    x≠012,…