I've looked up several related questions, but they do not answer what I am looking for. Please give link if this is a duplicate.

What I eventually want to know is why


converges to a finite value.

I mean, we have to know that the limit converges before we can call it a constant, right?

I know that $f(n)=\left(1+\dfrac1n\right)^n$ is strictly increasing, but how do we know it is bounded above?

Also, I tried using the binomial theorem to write

$$\lim_{n\to\infty}\left(1+\frac1n\right)^n=\lim_{n\to\infty}\left[\sum_{k=0}^n\frac{n!}{n^k k! (n-k)!}\right],\tag{1}$$

but I am unsure how to prove that the sum converges for $n\to\infty,$ because the general term depends on both $n$ and $k.$

Here is what I think will work.

Let $a_{n,k}=\dfrac{n!}{n^k k! (n-k)!}.$ Then $a_{n+1,k+1}=\dfrac{(n+1)!}{(n+1)^{k+1} (k+1)! (n-k)!}.$

Using the Ratio Test, we get

$$\lim_{n\to\infty\atop k\to\infty}\left|\frac{a_{n+1,k+1}}{a_{n,k}}\right|=\lim_{n\to\infty\atop k\to\infty}\left[\frac{1}{k+1}\left(\frac{n}{n+1}\right)^k\right].$$

where the limit on the RHS is less than 1.

That may take care of the case where $n\in\mathbb{N},$ so I suppose that (1) can be generalized using "Newton's generalised binomial theorem."

  • $\begingroup$ @martinsleziak, and how would I begin that? I don't see how your comment helps. $\endgroup$ Jun 22, 2015 at 14:10
  • $\begingroup$ My comment was posted in order to provide you (and other readers of the question) with links to some very similar questions that have been asked before on the site. I was able to find two of them, maybe there is much more than that. The links are now displayed also in the sidebar on the right among the linked questions. $\endgroup$ Jun 22, 2015 at 14:14
  • $\begingroup$ Sorry failed to see your comment was a link. $\endgroup$ Jun 22, 2015 at 14:15
  • $\begingroup$ Sorry failed to see your comment was a link. Of the two links, the main answers are a repeat of the one posted below. The one exception being the use of the Binomial Theorem in comparison to an expression that decomposes into partial fractions, which is not quite clear to me. $\endgroup$ Jun 22, 2015 at 14:20

3 Answers 3


$$\begin{align}\left(1+\frac 1n\right)^n&=\sum_{k=0}^{n}\binom{n}{k}\left(\frac 1n\right)^k\\&=\sum_{k=0}^{n}\frac{n(n-1)\cdots(n-k+1)}{k!}\left(\frac 1n\right)^k\\&=\sum_{k=0}^{n}\frac{1}{k!}\left(1-\frac 1n\right)\left(1-\frac 2n\right)\cdots\left(1-\frac{k-1}{n}\right)\\&\lt \sum_{k=0}^{n}\frac{1}{k!}\\&=1+\frac{1}{1}+\frac{1}{2}+\frac{1}{3\cdot 2}+\frac{1}{4\cdot 3\cdot 2}+\cdots+\frac{1}{n\cdot(n-1)\cdots 2}\\&\lt 1+\frac{1}{2^0}+\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots+\frac{1}{2^{n-1}}\\&=3-\frac{1}{2^{n-1}}\\&\lt 3\end{align}$$

  • 1
    $\begingroup$ Line 4 to line 5 didn't make sense to me at first glance, so I suggested an edit. Very nice! I want to let this simmer a bit to see how many different ways people come up with for the upper bound. Also, does the method I proposed in the question work? $\endgroup$ Jun 22, 2015 at 12:50
  • $\begingroup$ Old but useful. There is a lot left out. Line 4 is due to limit of products equaling the product of limits, and each factor aside from the first goes to 1. I'd have to think about the other line more. $\endgroup$ Jul 22, 2021 at 22:59

First you should know that: $$ (1) \ln(1+x)=x-\frac{x^2}{2}+O(x^3)$$ Also consider that ln is an injective function meaning if $\ln(A)=1$ then $A=e$
suppose $A=(1+\frac{1}{n})^n$ So: $$ \lim_{n\to\infty} \ln(A) = \lim_{n\to\infty} \ln((1+\frac{1}{n})^n = \lim_{n\to\infty} n\ln(1+\frac{1}{n}) $$ Applying (1): $$ \lim_{n\to\infty} n\ln(1+\frac{1}{n}) = \lim_{n\to\infty} n(\frac{1}{n}-\frac{1}{2n^2}+O(\frac{1}{n^3}))$$ $$\lim_{n\to\infty} n(\frac{1}{n}-\frac{1}{2n^2}+O(\frac{1}{n^3})) = \lim_{n\to\infty} (1-\frac{1}{2n}+nO(\frac{1}{n^3})) = 1 $$ Since $$ \lim_{n\to\infty} \ln(A) = 1 $$ Than $$ \lim_{n\to\infty} A = e $$ So e is the least upper bound.

  • $\begingroup$ Thanks. Still waiting for a comment on whether the method I proposed in the question will work. Seems to me it should! $\endgroup$ Mar 1, 2021 at 19:03

I do not understand how your approach should work. First, we need to show that $\sum_{k=0}^n a_{n,k}$ is bounded. How the fact that the double limit $\lim_{n\to\infty\atop k\to\infty}\frac{a_{n+1,k+1}}{a_{n,k}}$ is smaller than $1$ (if it exists) is helpful for this? Next, $f(n)$ is strictly increasing, so for any positive $x$, $f(x)$ is bounded from above by $f(n)$ for any natural $n>x$, so why do we need to generalise binomial theorem for noninteger $n$?

On the other hand,

$$\sum_{k=0}^n\frac{n!}{n^k k! (n-k)!}\le \sum_{k=0}^n\frac{n!}{n^k(n-k)!}\cdot \frac 1{k!}\le$$ $$ \sum_{k=0}^\infty \frac 1{k!}\le 2+\sum_{k=2}^\infty \frac 1{k(k-1)} =2+\sum_{k=2}^\infty \left(\frac 1{k-1}-\frac 1{k}\right)=3.$$

  • $\begingroup$ My point is simply to use the ratio test to attach to a related sum from binomial Theorem. It's not whether we NEED to it's a question of CAN WE? $\endgroup$ Oct 29, 2023 at 0:10

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