prove that.?

Question

2/3+2/5+....+2/2n+1=1 and
2(b-a)/(b+a)a>0

Answer 1

2/3 + 2/5 + ... + 2/(2n + 1) < ln(n + 1), for n >= 1.

To solve this, we use mathematical induction.

a) Base case: Let n = 1. Then
LHS = 2/3 ... 2/3 = 2/3 ~= 0.66
RHS = ln(1 + 1) = ln(2) ~= 0.69
LHS < RHS, so the formula holds true for n = 1.

b) Assume the formula holds true for up to n = k. That is,

2/3 + 2/5 + ... + 2/(2k + 1) < ln(k + 1)

Then

2/3 + 2/5 + ... + 2/(2(k + 1) + 1) is the same as

2/3 + 2/5 + ... + 2/(2k + 1) + 2/(2(k + 1) + 1)

We know that by our induction hypothesis, this is
<= ln(n) + 2/(2(k + 1) + 1)

Simplifying that,

ln(n) + 2/(2k + 3)

(Want to prove ln(n) + 2/(2k + 3) <= ln(n + 2))

I got stuck at this point.