Expert: Paul Klarreich - 10/5/2005

Question
Hello Paul,
I have a question on mathematical induction:

Prove by MI that:

((n+1)/n)^n <= 3    for all positive integers n

Thanks in advance  

Answer
Hi, Sobhi,

Sorry I don't have a proof for you.  The sequence is well known to have a limit, which is denoted 'e', and is generally considered the definition of e.

And I recall that this particular problem is part of the proof that the limit exists.  (A monotone increasing sequence that is bounded has a limit, so proving that the terms are all less than 3 proves the existence of a limit.)

Normally, you would start by:

1. Showing that S(1) is true; (2/1)^1 = 2 < 3

2. Assuming the statement true for n = k.

3. Using that fact to prove the statement for n = k + 1.

I'll keep looking at it.  I also sent it to a couple of well-known mathematicians.  [Actually they are my children; they are mathematicians and I know them well.]

Paul