3
$\begingroup$

Why the following is not correct? : $$\lim_{n \to \infty} (1+\frac1n)^n = \lim_{n \to \infty} (1+\frac1n) \lim_{n \to \infty} (1+\frac1n) \dots \lim_{n \to \infty} (1+\frac1n) = 1 \times 1 \times \dots 1 = 1.$$ I guess that the problem (as usual) lies in infinity, but if so how? I checked all the theorems in Fitzpatrick's book of Calculus, nothing prevents to do the limits infinitely many times (esp. for a convergent one)!

  • 0
    You have an indeterminate form. I guarantee you if you search, you'll find many questions on this particular limit – 2017-02-22
  • 1
    Because $1^{\infty}$ is indeterminate – 2017-02-22
  • 0
    See [here](http://math.stackexchange.com/questions/2067849/why-does-lim-n-to-infty1-frac1nn-e-instead-of-1), [here](http://math.stackexchange.com/questions/136784/why-lim-limits-n-to-infty-left1-frac1n-rightn-doesnt-evaluate-to), or [here](http://math.stackexchange.com/questions/10490/why-is-1-infty-considered-to-be-an-indeterminate-form). – 2017-02-22
  • 0
    To add on to Noah's already good hint: at $\infty$, you end up taking a limit with infinitely many terms. Your rules with distributing limits ONLY work for finitely many terms. – 2017-02-22
  • 0
    Personally, I always thought it was amazing that this limit exists at *all*, much less that it isn't even sort of $1$. – 2017-02-22

3 Answers 3

7

Well one major problem is that you are, in a sense, taking the limit before taking the limit. How many terms are you multiplying?

4

You have written:

$$\lim_{n \to \infty} (1+\frac1n)^n = \lim_{n \to \infty} (1+\frac1n) \lim_{n \to \infty} (1+\frac1n) \dots \lim_{n \to \infty} (1+\frac1n) = 1 \times 1 \times \dots 1 = 1,$$

But this is not correct, because it suggests a finite and unchanging number of quantities being multiplied together. As $n$ gets bigger, you multiply more. When $n$ is 20, for example, you have $(1+0.05)^{20}\approx2.65$, already. As you take away from the added term, you multiply more. And for no value of $n$ do you have that $\left(1+\frac{1}{n}\right)=1$.

Does this help?

1

In your first equality it seems that you equating two very different expressions:

$$ \lim_{n\to\infty}\left[ \left( 1+\frac{1}{n}\right)^n\right] \text{and }\left[\lim_{n\to\infty} \left( 1+\frac{1}{n}\right)\right]^n$$

which are not equal. For a general expression $p(n)$,

$$\lim_{n\to\infty} \left[p(n)^n\right] \ne \left[\lim_{n\to\infty}p(n)\right]^n.$$

While this may be true for some $p(n)$, it must be shown to be true first before equality is taken.