Proving an Inequality: How to Use Induction to Show a Sum is Less Than 3

[silvermane]

Homework Statement

Prove that

2 \leq 1+ \sum(m=1 to n) 1/m! \leq 1 + \sum (m=1 to n) (1/(2^(m-1))) < 3

The Attempt at a Solution

I've proved by induction that 2^m-1 \leq m!, so it just follows that
1 + (1/(2 ^ (m-1))) \geq 1 + (1/m!), and their sums are the same inequality.

After this however, I'm having issues proving the rest. Any hints or tips are greatly appreciated!

Thanks in advance!

 

[micromass]

The last inequality is just the sum of a geometric series. They've got formulas for that...
The first inequality is quite easy, just take the first term of the sum and you've got it already...

 

[silvermane]

The last inequality is just the sum of a geometric series. They've got formulas for that...
The first inequality is quite easy, just take the first term of the sum and you've got it already...

We're not supposed to use formulas from calc two. We must use analysis, otherwise I would have solved it. :(

 

[Hurkyl]

It's not a formula from calc 2, it's a formula from your high school algebra classes. If nothing else, you could just repeat the derivation if you really want to redo arithmetic from first principles.

 

[╔(σ_σ)╝]

Does the question say not to use formulas? The reason I asked is that in my analysis course we took the geometric series formula as given.

Technically, you can derive the formula if you are not. allowed to just pull it out of your hat.

If not, you want to consider something else perhaps maybe induction. You would probably meet a few road blocks with induction though.

AM- GM may help also but I haven't tried it out myself.


EDIT

hurkyl beat me to it lol.

 

[silvermane]

Yeah, the question says not to use any formulas. That's why I was so confused. I have the solution with the formula, but I'm also trying to work one out soley from induction and I'm just having so much trouble with it :(

Thanks for helping everyone! :)
Also, icystrike, what you've done helped me finish some other thoughts on what I have down thusfar as well. Thanks!

 

[╔(σ_σ)╝]

Well repeat the derievation of the formula in that case. Strictly speaking, it is not using formulas!

The problem with induction is that you do not have a formula on the other side, you have a 3. So even if you proved that for n-1 the statement is true you have no way of showing it is true for n.