Exercises 4.2 ove that the sequence (1 + z/n)"; n = 1, 2, 3,..., converges uni-...
bn converges 18. Let (an)n=1 and (bn)n=1 be sequences in R. Show that if and lan – an+1 < oo, then anbr converges.
(3) Prove that the sequence fn (x(max10,z - n))2 does not converge uniformly on IR, but converges uniformly on compact subsets of R (3) Prove that the sequence fn (x(max10,z - n))2 does not converge uniformly on IR, but converges uniformly on compact subsets of R
Determine if the sequence {an}n=1 3n3 with an = converges. If it converges write its /2n6 + 6n2 limite if it doesn't converge write diverge. Answer:
in positive sense (both exercises) 1) 2)where z2 is the root of z^3 - 1 that is in the second quadrant. -0<R< 3 1 dz. 12-11=R 23 -1 Jos os -0907 1 dz 12-22=0.997 23 - 1
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
Show that a bounded and monotone sequence converges. Here a sequence is called monotone, if it is either monotone increasing, that is for all or monotone decreasing, in which case for all . in Sn=1 An+1 > an neN an+1 < an We were unable to transcribe this image
x[n] = { Consider the discrete sequence S (0.5)" 0<n<N-1 otherwise a) Determine the z-transform X(2)! b) Determine and plot the poles and zeros of X(2) when N = 8!
Given that the sequence defined by - 1 2+1 = 5-1 an is increasing and an < 5 for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that {z.) is Cauchy. Is this result true under the condition Irn +1-Fml < rt Let xi = 1 and xn +1 = (Zn + 1)/3 for all n e N. Find the first five terms in this sequence. Use induction to show that rn > 1/2 for all n and find the limit N. Prove that this sequence is non-increasing, convergent,
(h) Define f : [0, 2] + R by 122 if 0 <<<1 f(x) = { ifl<152 Using the limit definition of the derivative and the sequence definition of the limit prove that f'(1) does not exist.