Question

Prove for any natural number
1/(n + 1) + 1/(n + 2) + 1/(n + 3) + · · · + 1/2n > 13/24

Answer 1

I think at first I need to prove that it is true for n=1
But if n=1 the first number is 1/2 and the last number also half.
Since the series is decreasing it is impossible to have the last number = the first number. Hence there is only 1 number in the series -> 1/2
But 1/2 is greater than 13/24

Answer 2

This is not true for n = 1; so I'll start the induction at n = 2.

Base Case:
This is true for n = 2, because 1/2 + 1/3 + 1/4 = 26/24 > 13/24.

Inductive Step:
Assuming that 1/(n+1) + 1/(n+2) + 1/(n+3) +.....+ 1/(2n) > 13/24:

1/(n+1) + 1/(n+2) + 1/(n+3) +.....+ 1/(2n) + 1/(2n+1) + 1/(2n+2)
> [1/(n+1) + 1/(n+2) + 1/(n+3) +.....+ 1/(2n)] + 1/(2n+1) + 1/(2n+2)
> 13/24 + 1/(2n+1) + 1/(2n+2), by inductive hypothesis
> 13/24 + 0
= 13/24, completing the inductive step.
similar- http://answers.yahoo.com/question/index?qid=20110909091202AA0lbEY