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

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In summary, the student is trying to prove that the statement "for every n there is a m-1 such that 1 + (1/(2 ^ (m-1))) \geq 1 + (1/m!), and their sums are the same inequality" is true. They have proved it for n-1, but are having trouble proving it for n. They are also looking for help with induction.
  • #1
silvermane
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Homework Statement


Prove that

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


The Attempt at a Solution



I've proved by induction that 2m-1 [tex]\leq[/tex] m!, so it just follows that
1 + (1/(2 ^ (m-1))) [tex]\geq[/tex] 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! :blushing:
 
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  • #2
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...
 
  • #3
micromass said:
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. :(
 
  • #4
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.
 
  • #5
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.
 
  • #6
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!:blushing:
 
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  • #7
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.
 

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

What is the purpose of proving an inequality?

The purpose of proving an inequality is to demonstrate that a certain relationship between two quantities is true. This can be useful in various fields of science and mathematics, as it allows for the validation and comparison of different theories and hypotheses.

How do you prove an inequality?

To prove an inequality, you need to show that one side of the inequality is always less than or greater than the other side. This can be done through various methods, such as algebraic manipulation, substitution, or mathematical induction.

What are some common techniques used in proving inequalities?

Some common techniques used in proving inequalities include: using the properties of inequalities (such as the transitive, symmetric, and reflexive properties), using basic algebraic operations (such as addition, subtraction, multiplication, and division), and using special inequality rules (such as the triangle inequality or the Cauchy-Schwarz inequality).

Can inequalities be proven for all values?

Inequalities can be proven for all values as long as certain assumptions and restrictions are met. For example, if the inequality involves real numbers, then it can be proven for all real numbers. However, if the inequality involves complex numbers, then it can only be proven for complex numbers within a certain range or set of conditions.

What are some applications of proving inequalities in science?

Proving inequalities has many applications in science, including in physics, chemistry, economics, and biology. For example, in physics, inequalities can be used to determine the stability and efficiency of physical systems, while in economics, they can be used to analyze market trends and predict future outcomes.

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