# Proof of this limit formula for e

• Arnoldjavs3
In summary, the student is trying to do a proof for the following equation: $$lnL = \frac{ln(1+1/x)}{x}$$However, they get stuck on the first step and end up with a tautology. The second step is to show that for any n expansion, even as n ⇒ ∞, the sum is always less than 3. This leads to a limit.
Arnoldjavs3
Poster warned for not typing in the problem statement

## Homework Statement

http://prntscr.com/dcfe0u

## The Attempt at a Solution

So I'm not really strong in proofs but I think you may be able to do something like this:$$lnL = \frac{ln(1+1/x)}{x}$$
$$lnL = \frac{1/x^2}{1+1/x}$$
and then more simplifying I get something like:

$$lnL = \frac{x}{x^2+x^3}$$

At this point I think I'm just incredibly off. If I were to continue that I think I would have to multiply the right side by f(x) to remove ln from the Limit.

How do I approach this proof?

I can't quite follow your equations. I don't think they are correct. But to start with, what is ##e##? And how is the limit defined?

The proof will depend on how your text materials define ##e##.

Edit your post and include the definition of ##e## in the "relevant equations" section.

What definition of ##e## are you using?

Arnoldjavs3 said:

## Homework Statement

http://prntscr.com/dcfe0u

## The Attempt at a Solution

So I'm not really strong in proofs but I think you may be able to do something like this:$$lnL = \frac{ln(1+1/x)}{x}$$
The above is incorrect. Starting with
##L = (1 + \frac 1 x)^x##
Taking the natural log of both sides yields this:
##\ln L = x \ln (1 + \frac 1 x)##
That's different from what you have above.
Arnoldjavs3 said:
$$lnL = \frac{1/x^2}{1+1/x}$$
and then more simplifying I get something like:

$$lnL = \frac{x}{x^2+x^3}$$

At this point I think I'm just incredibly off. If I were to continue that I think I would have to multiply the right side by f(x) to remove ln from the Limit.

How do I approach this proof?

After a bit of googling I came up with a solution I think it's right(very different from the jibberish i had in OP) but I'm not sure.

I did the solution on mywhiteboard as this seems very tedious to do on mathjax.

http://prntscr.com/dchmps

After a second look that looks messy. Sorry about that lol

Arnoldjavs3 said:
After a bit of googling I came up with a solution I think it's right(very different from the jibberish i had in OP) but I'm not sure.

I did the solution on mywhiteboard as this seems very tedious to do on mathjax.

http://prntscr.com/dchmps

After a second look that looks messy. Sorry about that lol
Not only that, it doesn't make much sense. You have a lot of stuff in there that is either meaningless (such as ##\lim_{x \to \infty}(1 + 0)^{\infty}##) or unrelated to this problem (such as the definition of the derivative you have in the middle).

If you continue from what I have in post #5, you should be able to finish it off. The key idea is that ##\lim \ln f(x) = \ln \lim f(x)##, as long as f is continuous. IOW, you can switch the order of the limit and log operations, under suitable conditions.

Arnoldjavs3 said:
After a bit of googling I came up with a solution I think it's right(very different from the jibberish i had in OP) but I'm not sure.

I did the solution on mywhiteboard as this seems very tedious to do on mathjax.

http://prntscr.com/dchmps

After a second look that looks messy. Sorry about that lol

Please type up your work; many of us will not open attachments or look at picture files. (Please read the post "Guidelines ford students and helpers", by Vela, for more information on this issue.)

$$\lim_{x\rightarrow\infty}\left(1+\frac{1}{x}\right)^x%=\exp\left(\lim_{x\rightarrow\infty}\frac{\log\left(1+\frac{1}{x}\right)}{\frac{1}{x}}\right)$$
use
continuity of exponential
$$\lim_{x\rightarrow\infty}e^{f(x)}=\exp\left(\lim_{x\rightarrow\infty}f(x)\right)$$
exponential identiy
$$x^y=\exp(y \log(x))$$
value of e
$$e=\exp(1)$$
and derivative of log
$$\lim_{x\rightarrow\infty}\frac{\log\left(1+\frac{1}{x}\right)}{\frac{1}{x}}=\log^\prime(1)=1$$

The above "solutions" are all tautologies. They all assume that e exists and is the base of natural logarithms. Now, this is true, but the proof of this is outlined at
http://planetmath.org/convergenceofthesequence11nn, I have found no other site with this proof.

Essentially done in two steps.

1) (1 + 1/n)^n is expanded (binomial expansion) to n and n + 1 terms
2) comparing term by term, the n + 1 expansion is ALWAYS greater than the n expansion
3) thus the sum of the series is monotone increasing

4) the second step is to show that for any n expansion, even as n ⇒ ∞, the sum is always less than 3. Hence the sum of the series is bounded. Monotone increasing with an upper bound ⇒ a limit.

The calculation for e would be either a Taylor series or the use of an Excel spreadsheet. With the latter, one would, through iteration, obtain results until the results do not change, which would be the limit of the expansion that Excel could produce.

That is a more rigorous proof and does not require one to use a tautology to come up with an answer.

## What is the proof of the limit formula for e?

The proof of the limit formula for e involves using the definition of the number e as the limit of (1 + 1/n)^n as n approaches infinity. This definition is equivalent to the limit formula for e, which is lim (1 + x)^1/x as x approaches 0. By using mathematical manipulations and theorems, such as the binomial theorem and L'Hopital's rule, it can be shown that these two limits are equal.

## Why is it important to prove the limit formula for e?

Proving the limit formula for e is important because it is the foundation for many other important mathematical concepts and formulas involving the number e. It also helps to deepen our understanding of the nature of e as a mathematical constant and its relationship to exponential functions.

## What are the steps involved in proving the limit formula for e?

The steps involved in proving the limit formula for e include defining the number e as the limit of (1 + 1/n)^n as n approaches infinity, using the binomial theorem to expand (1 + 1/n)^n, applying L'Hopital's rule to evaluate the limit of the expanded expression, and simplifying the resulting expression to match the limit formula for e.

## Are there any alternative proofs for the limit formula for e?

Yes, there are alternative proofs for the limit formula for e, such as using Taylor series or using the definition of the derivative of e^x to show that the derivative of e^x is equal to e^x itself. However, the most commonly used and accepted proof is the one using the definition of e as the limit of (1 + 1/n)^n.

## How can understanding the proof of the limit formula for e be applied in real-world situations?

Understanding the proof of the limit formula for e can be applied in various real-world situations, such as in finance, physics, and engineering. It can be used to model exponential growth and decay, calculate compound interest, and solve differential equations involving exponential functions. It also has applications in statistics, probability, and signal processing.

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