Problem on sequences and series and limits and convergence

Question

Assignment:

The sequence S_­n­ = ((1+ (1/n))^n converges, and its limit can be used to define e.

a) For a fixed integer n>0, let f(x) = (n+1)xⁿ - nxⁿ⁺¹ . For x >1, show f is decreasing and that f(x) . Hence, for x >1;

Xⁿ(n+1-nx) < 1

b) Substitute the following x-value into the inequality from part (a)

X= ((1+(1/n))/((1+(1/(n+1))

And show that:

x²((n+1)/(n+2)) < 1

c) Use the inequality from part (b) to show that s_n­ < s_n+1­ for all n > 0. Conclude the sequence is increasing.

d) Substitute x= 1 +(1/2n) into the inequality from part (a) to show that

((1+ (1/2n))ⁿ < 2.

e) Use the inequality from part (b) to show that s_2n <4. Conclude the sequence is bounded.

f) Use parts (c ) and (e) to show the sequence has a limit.

Provide complete and step by step solution for the question and show calculations and use formulas.