# Homework Help: Prove that ?

1. Feb 18, 2012

### topengonzo

1. The problem statement, all variables and given/known data
Prove that 1+1+1/2!+1/3!+....+1/n! < 3

2. Relevant equations
None

3. The attempt at a solution
Any suggestion how can I start?
I dont want the solution that 1+1+1/2!+1/3!+...+1/n!=(1+1/n)^n and lim (1+1/n)^n = e<3 since the 2nd part is to prove (1+1/n)^n <3 . I don't want a solution. I just want to know how to start with it.

2. Feb 18, 2012

### tiny-tim

welcome to pf!

hi topengonzo! welcome to pf!

hint: can you see a way to prove that 1/2!+1/3!+....+1/n! < 1 ?

3. Feb 18, 2012

### topengonzo

hi tiny-tim

this is the first thing i thought of and i actually worked without the first 2 terms. Prove by induction on this problem is impossible i think. So i dont have a clue how to solve it. :S

4. Feb 18, 2012

### tiny-tim

1/2+1/2.3+1/2.3.4+....+ < ?

5. Feb 18, 2012

### sunjin09

hint: does the original series look familiar to you at all? Have you compared it to any series you already know?
(If you haven't learned taylor expansion ignore what I said here)

6. Feb 18, 2012

### topengonzo

i dont get what u mean. I am thinking of proving that sum of next terms is less than equal to current term that is:
(1/3!) + (1/4!)+(1/5!)+....(1/infinity!)<(1/2!)
(1/4!) + (1/5!)+(1/6!)+....(1/infinity!)<(1/3!)
.... (1/(n+1)!) + (1/(n+2)!)+.... (1/infinity!)<(1/n!)

Should I reach a solution with this method or can I solve it this way?

7. Feb 18, 2012

### topengonzo

I think its taylor expansion will give me e<3 and solved. Is there another way to solve it?

8. Feb 18, 2012

### sunjin09

deleted

Last edited: Feb 18, 2012
9. Feb 18, 2012

### tiny-tim

sunjin09, please don't give the full answer

10. Feb 18, 2012

### topengonzo

THANK YOU VERY VERY MUCH!

So I prove 1/k! < 1/ 2^(k-1) for k>=3 by induction and then i can say my series < 1+ 1 + 1/2 + 1/4 + 1/8 + ... which is geometric series with r=1/2 and first term 1
implies 1+ 1/(1-1/2) = 3

11. Feb 18, 2012

### checkitagain

That is the same as $\dfrac{1}{2} \ + \ \dfrac{1}{2}\cdot 3 \ + \ \dfrac{1}{2}\cdot3\cdot4 \ \ + \ ... \ + \ < \ ?$

(You're using the dots for multiplication. Written out horizontally,
grouping symbols are needed.)

Instead, it could be correctly shown horizontally as:

1/2 + 1/(2*3) + 1/(2*3*4) + ... + < ?

or as

$1/2 \ + \ 1/(2\cdot 3) \ + \ 1/(2\cdot3\cdot4) \ + \ ... \ + \ < \ ?$

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