(f) Assume that g(x) > bna" is convergent on (-R, R). Instead of assuming that f(rn) = 0 assume that f(rn) = g(xn) for all n. Show that f(x) = g(x) on (-R, R). %3D

Prove that the sequence {cn} converges to c if and only if the sequence {cn- c} converges to 0.

I had tried Xn=(-1)^nbut since 2((-1)^n)^n =2(-1)^(n^2) (which is not always 1) it doesn't converge to some real number.

Let f(r) = > anx" converges on an interval (-R, R) and let In be a non zero sequence in n=0 (-R, R) that converges to 0. If f(xn) = 0 for every n show that

Let {a,}1 be a convergent sequence defined by a1 = V5 and an+1 = V5a,. Find lim a,.

Write out the first five terms of the sequence , determine whether the sequence converges, and if so find its limit. [(-1)*-1 ¯ (n+4) . n=1 Enter the following information for an = (-1)"-1 (n+4)² · = Ip a2 = аз a4 A5 = (-1)"–1 lim (n + 4)² Does the sequence converge

Q\ Find the infimum and supremum of the sequence (a„)nen = (n(1+ (-1)") + n-'((-1)" – 1). NĒN

Th 2n +2 Diverges to - Converges to 1. Converges to 8. Diverges to +a0

If x1, y1, are two positive unequal numbers and xn = (xn-1+ yn-1)/2 and yn= sqrt(xn-1yn-1)  for all n>=2, Prove that the sequences <xn> and <yn> are monotonic and they converge to the same limit. *Prove each step

1) In order to use the integral test, we must verify that the sequence is decreasing (and positive and continuous). To determine that a sequence is decreasing, one of the the method is to treat the sequence as a function and use calculus to see if the first derivative is negative or not. For the following sequence, use the method with calculus to verify that it is decreasing: a) -n пе 1 b) n In n n=2