(d) For each N > 0, there is ɛ > 0 such that an - a| < N-¹ when n > N +ɛ.

Q: Theorem 64. Let {pk}1 converge to point If y # x, then {pk}1 does not converge to y. x.

A: Given that sequence pkk=1∞ converges to point x.

Q: Show that it is not true that for every error 1/m there exists n such that |Ek=1 polt – r/(1 - r)| <…

A: We will show that for every m there does not exist any unique n that works for all r in (0,1)

Q: 4 Suppose that lim anti n7708 an =P -=p. For which values of P must Σ2" an converge?

A: To determine the values of p  for which the series ∑n=1∞​2nan​  converges, we can use the given…

Q: Show that the following sequence, {an}, 'n = 1 %3D converges where Vn = 1 - п! an - using Monotone…

A:

Q: 2. Suppose that {an}1 converges to 1 and {bn}1 converges to 2. For n € N, define an: if n is odd; Cu…

A: Please see the below picture for detailed solution.

Q: is it divergent, convergent or absolutely convergent?

A: We have given a series ,  ∑k=0∞ sin2k+1π2 k3k+1! Here , sin2k + 1π2 = sinkπ + π2 sin2k + 1π2  =…

Q: 5. Show that the following sequence Pa - n be before pn-p| ≤5 x 10-²? 1 2n converges linearly to p =…

A:

Q: an2

A:

Q: x +1 29. Find the values of x for which > converges. 4 n= a) x=tV3 only b) x=1 only c) -3V3 d) x

A: we have to find value of x

Q: 8. 2. Let an be convergent with an > 0 for all n. Now consider (an). n=1 n=1 a) Explain in words why…

A: a) Given that series an is convergent with an &gt; 0.

Q: True or false: if (x) converges, then every subsequence (x) is Cauchy. n=] k=1

A: