3 40 points Give examples or explain why no example is possi

3. (40 points) Give examples or explain why no example is possible. Explain how your example fits or why no example is possible

Solution

3. (e) Let, x_n = (1 + 1/n)ⁿ

x_n is rational number for all integer n.

Then the sequence {x_n} of rational numbers converges to an irrational number e.

Lim (1 + 1/n)ⁿ = e as n (This is a standard result)

(f) Let, f(x) = sin (1/x) for all x (0, 1]

Then the function f(x) is continuous, but not uniformly continuous from (0, 1] into R.

Proof:

Let us assume that f is uniformly continuous on (0, 1]. Then for every Cauchy sequence {x_n} in (0, 1], the sequence {f(x_n)} must be a Cauchy sequence in R. Let us consider the sequence {x_n} where x_n = 2/(n), n N. This is a Cauchy sequence in (0, 1]. The sequence {f(x_n)} is {1, 0, -1, 0, … …}. This is not a Cauchy sequence in R. Therefore, f is not uniformly continuous on (0, 1].

(g) A power series having infinite radius of convergence is xⁿ/n!

Proof:

Let r be the radius of convergence of the power series xⁿ/n!

Then, 1/r = lim_n |n!/(n+1)!| = lim_n {1/(n+1)}

Therefore, r = lim_n (n+1) =

Formula for Radius of convergence r of a power series a_nxⁿ is given by 1/r = lim_n |a_n+1/a_n|