Question

We show that the sequence converges; 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) < 1. Hence, for x > 1.




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



and show that




(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




(e)
Use the inequality from part (d) to show s2n < 4. Conclude the sequence is bounded.

(f)
U

Answer 1

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

Answer 2

any ideas whatsoever?

Answer 3

The second term...is strange..-nxn+1? Is that -xn^2+1?

Answer 4

-nx^n+1

Answer 5

-nx^(n+1) or -nx^(n) +1?

Answer 6

-nx^(n+1)