In the previous part, we talked about some limits in the limit theory that are considered as wonderful. If you haven’t read it yet, let me give you a quick brief. The term “The Wonderful Limits” is widely known in post-soviet countries, and there are two such limits. The second one is:
This limit is an introduction to the e number. You know, the exponent is a “magic” number in maths, and this is the reason why this limit is also called the wonderful one.
Actually, we can not just prove that the limit equals the exponent because the exponent is the value of this limit by definition. So here, we need to prove that the sequence (1 + 1/x)ˣ has a limit when x approaches infinity. This proof is more complicated than the previous one.
To prove the statement, we’ll use “The Convergence of a Monotone Bounded Sequence of Real Numbers Theorem” for an increasing sequence. It states the following:
If a sequence of real numbers is increasing and bounded above, then its supremum is the limit.
The Case of the Natural Numbers
First, let’s prove that the sequence (1 + 1/n)ⁿ is limited when n approaches infinity and n is a natural number. It’ll help us to get the whole proof.
By the Binomial theorem, we have:
As you might already know, when n grows, 1/n decreases. Hence, if we take the opposite values, the sequence (1–1/n), (1–2/n) … is becoming increasing. That’s why (1 + 1/n)ⁿ is an increasing sequence and, therefore, monotone. So, we’ve just proved that the sequence satisfies the first part of the theorem.
Now, we need to show that this sequence is bounded:
By using the geometric progression sum formula, we get the next inequality for the second component of the right side of the inequality above:
So, after summing up the parts of the expression, we get the upper bound of the sequence:
And here we are! The sequence is located between 2 and 3 on the real line. So, it’s bounded and monotone ⇒ it has a limit (let’s call it e).
The Case of the Real Numbers
This one is kinda tricky. We need to consider two situations: when x approaches +∞ and –∞.
x approaches +∞
You know that any positive real number is located between two positive integers (or 0). For instance, 37.55435 is located between 37 and 38. In this example, 37 is the whole part of our number. Generally, we write it as [x]. So, x always satisfies the next inequality: [x] ≤ x ≤ [x] + 1. That’s why we can claim that:
Thus, we get that:
We know the limit of (1 + 1/n)ⁿ, let’s use it!
By the squeeze theorem, we get the proof of our statement when x approaches + infinity.
x approaches –∞
Here, let’s substitute -x with t. So, we get:
The Conclusion
Finally, here we are! It is a really long proof, and don’t trust websites that try to prove it in a short way by using natural logarithms and the exponent itself. This limit is the introduction to the exponent, and we need to act like we know nothing about this “mysterious” number.
There are a lot of reasons why mathematicians like the exponent this much, but it’s a topic for another article. But this limit is wonderful because it introduces this number to us.
I hope you’ve enjoyed reading this proof (and the first part as well)! See you in the next mathematical articles!
