How would you prove this little inequality?

  • Thread starter stunner5000pt
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In summary, the conversation discusses a problem with proving the inequality (1+ 1/n)^n <= e <= (1+1/n)^n+1 without using power series for e. A hint is given about the limit of (1+ 1/n)^n. It is suggested that the proof can be done by reasoning that (1+1/n)^n will always be less than (1+1/n)^n * (1+1/n) until n approaches infinity.
  • #1
stunner5000pt
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i'm stuck trying to prove this little inequality:

(1+ 1/n)^n <= e <= (1+1/n)^n+1
is there a way to prove this without without resorting to power series for e (because we're not allowed to, and we don't know this yet) and also note that n is a natural number, (positive integer).
 
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  • #2
Here's a hint:

[tex]\lim_{n \to \infty} \Big( 1 + \frac{1}{n} \Big)^n = e[/tex]

cookiemonster
 
  • #3
ummmm

ok i know tha already i just don't know how to prove it give me hint on hwo to prove it
 
  • #4
Just how formally do you want to prove it?

It's pretty easy to notice that for [itex]n<\infty[/itex], the left side is less than e. When [itex]n = \infty[/itex], it is exactly equal to e.

The same holds true for the right side, except that it's always greater than e except when [itex]n = \infty[/itex].

cookiemonster
 
  • #5
You do it by power series for (1+x)^n valid when |x|<1 (ie x=1/n)
 
  • #6
Originally posted by stunner5000pt
i'm stuck trying to prove this little inequality:

(1+ 1/n)^n <= e <= (1+1/n)^n+1
is there a way to prove this without without resorting to power series for e (because we're not allowed to, and we don't know this yet) and also note that n is a natural number, (positive integer).
Why can you not just reason that (1+1/n)^n has to be less than (1+1/n)^n * (1+1/n), as, until n => infinity, 1+1/n will always be a positive value above * a positive value that will always make the right hand larger.
 

1. How do you define "proof" in the context of an inequality?

In mathematics, a proof is a logical argument that demonstrates the truth or validity of a statement. In the context of inequalities, a proof involves using mathematical techniques to show that one side of the inequality is always greater than or less than the other side, depending on the direction of the inequality.

2. What are the different types of proofs that can be used for inequalities?

There are several types of proofs that can be used for inequalities, including direct proof, proof by contradiction, and proof by induction. A direct proof involves using known mathematical properties or theorems to show that the inequality is true. A proof by contradiction assumes that the inequality is false and then shows that this leads to a contradiction. Proof by induction is a method of proving inequalities for all natural numbers by first proving for a base case and then showing that if it is true for one number, it is true for the next number.

3. What is the importance of using mathematical notation in a proof of an inequality?

Mathematical notation is important in a proof of an inequality because it allows for clear and concise communication of mathematical ideas. Notation also helps to make the proof more rigorous and less reliant on language, which can be ambiguous. It also allows others to easily understand and follow the steps of the proof.

4. How do you determine which mathematical techniques to use when proving an inequality?

The choice of mathematical techniques to use when proving an inequality depends on the specific inequality being considered. It is important to carefully examine the inequality and identify any patterns or relationships between the terms. This can help determine if a direct proof, proof by contradiction, or proof by induction would be most appropriate. Additionally, knowledge of mathematical properties and theorems can guide the selection of techniques to use.

5. Can you provide an example of a proof for a simple inequality?

Yes, for example, to prove that for any real numbers a and b, if a < b, then a2 < b2, we can use a direct proof. We know that for any real number x, x2 > 0. So, if a < b, then a2 < ab and b2 > ab. Combining these two inequalities, we get a2 < b2, which proves the original inequality.

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