The sequence Sn = ((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)
And show that:
x2((n+1)/(n+2)) < 1
c) Use the inequality from part (b) to show that sn < sn+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 s2n <4. Conclude the sequence is bounded.
f) Use parts (c ) and (e) to show the sequence has a limit.
Please see the attached file for the fully formatted problems.
Increasing, Bounded, Sequences and Series, Limits and Convergence are investigated. The solution is detailed and well presented.