Exponential Series Proof

Date: 05/05/2001 at 22:55:22
From: Jake
Subject: Exponents/Algebra

Hi, 

I'm having trouble with an exponential question. 

In previous parts of the question, they give it that e^x is greater 
than or equal to 1 + x for all real values of x. They also tell you 
that:

     (1+1)(1+(1/2))(1+(1/3)) ... (1+(1/n)) = n+1

The question is: Using these two pieces of information, prove that:

     e^(1 + (1/2) + (1/3) + ... + (1/n)) > n

and also,

Find a value of n for which:

     1 = (1/2) + (1/3) + ... + (1/n) > 100

Thanks!

Date: 05/06/2001 at 17:18:24
From: Doctor Jaffee
Subject: Re: Exponents/Algebra

Hi Jake,

Here is how I would prove that e^(1 + (1/2) + (1/3) + ... + (1/n)) > 
n.

Since e^x is greater than or equal to 1 + x, it follows that:

   e^1 is greater than or equal to 1 + 1,
   e^(1/2) is greater than or equal to 1 + 1/2,
   e^(1/3) is greater than or equal to 1 + 1/3
    .
    .
    .
   e^(1/n) is greater than or equal to 1 + 1/n.

So, the product of (e^1)(e^(1/2))...(e^(1/n)) is greater than
(1 + 1)(1 + 1/2)(1 + 1/3)...(1 + 1/n)

Now, you can use the rule for multiplying exponential numbers to 
transform the left side into what you want, and the right side you 
know equals n + 1. You should be able to work it from there.

I'll give you a hint for the second problem. You know that 1/3 is 
greater than 1/4, so 1/3 + 1/4 must be greater than 2/4 or 1/2.  
Likewise, 1/5, 1/6, and 1/7 are all greater than 1/8, so the sum of 
the four of them must be greater than 4/8 or 1/2. In a like fashion, 
the sum of the next eight fractions is greater than 1/2, the sum of 
the sixteen fractions that follow is greater than 1/2, etc. You should 
be able to see how many 1/2's you need to exceed 100.

Give these problems a try and if you want to check your answers with 
me, write back. If you are having difficulties, let me know and show 
me what you have done so far, and I'll try to help you some more.

Good luck.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/11/2001 at 04:42:58
From: Jake
Subject: Re: Exponents/Algebra

Thank you so much! It helped a lot.

About the second problem, I understand your instructions but could you 
do it like this:

They ask you to find 1+ 1/2 + 1/3 +...+1/n > 100. Since they tell you 
that e^(1+1/2+1/3+...1/n) > 100 in the previous question, does it make 
sense just to say that it implies that any n > e^100 will do?

From: Doctor Jaffee
Subject: Re: Exponents/Algebra

Hi,

Yes, that makes perfectly good sense to me. What I wrote you before 
was correct, but it occurred to me later that a better approach would 
be to take the natural logarithm of both sides of

     e^(1+(1/2)+(1/3)+...+(1/n)) > n

then set ln n = 100. I assume that you ended up doing something 
similar to that to arrive at your conclusion.

I'm glad I was able to help you; I enjoyed working on the problem and 
I hope to receive more interesting questions from you.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/