Finding the limit of $\left(\frac{n}{n+1}\right)^n$
Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$
I'm pretty sure it goes to zero since $(n+1)^n > n^n$ but when I input large numbers it goes to $0.36$.
Also, when factoring: $$n^{1/n}\left(\frac{1}{1+\frac1n}\right)^n$$ it looks like it goes to $1$.
So how can I find this limit?
$$\lim_{n\to \infty}\left(\frac{n}{n+1}\right)^n = \lim_{n\to \infty}\frac {1}{\left(\frac{n+1}{n}\right)^n}=\lim_{n\to \infty}\frac {1}{\left(\frac nn+\frac{1}{n}\right)^n}=\lim_{n\to \infty}\frac {1}{\left(1+\frac{1}{n}\right)^n}$$ We know that: $$\lim_{n\to \infty}\left(1+\frac{1}{n}\right)^n=e$$ $$\color{red}{\boxed{\displaystyle\color{black}{\quad\therefore\quad\lim_{n\to \infty}\left(\frac{n}{n+1}\right)^n=\frac 1e\quad}}}$$
And for the intuitive part: $$\begin{array}{|c|c|} \hline \text{Value of }\color{black}{n} & \left(\frac{\color{black}{n}}{\color{black}{n}+1}\right)^{\color{black}{n}} \\ \hline 1 & 1.\overline6 \\ 2 & 0.562 \\ 3 & 0.512 \\ 4 & 0.482 \\ 5 & 0.462 \\ 6 & 0.448 \\ 7 & 0.438 \\ 8 & 0.430 \\ 9 & 0.424 \\ 100 & 0.373 \\ \hline \end{array}$$ And: $$\frac 1e\approx0.3678794...$$ Sure enough, we are approaching $1/e$ as $n$ tends to $\infty$.
Furthermore, you could think of it that way:
As $n$ gets really really large, $1/n$ will be very very small.
As $n$ gets really really large, the coefficients and number of terms of $(a+b)^n$ will also get really large. That's why even if $1/n\approx0$ when $n$ gets big, the expression $\left(1+\frac{1}{n}\right)^n$ will approach a number greater than one, and who has to do with the amount of terms and coefficients present in $(a+b)^n$, and that number is simply $e$. To show more what I mean, look at this example: $$\begin{align}(1+\color{blue}{0.1})^{10}=&1^{10}+\color{green}{10} \cdot1^9 (\color{blue}{0.1})+\color{green}{45}\cdot1^8 (\color{blue}{0.1})^2+\color{green}{120} \cdot1^7 (\color{blue}{0.1})^3+\color{green}{210} \cdot1^6 (\color{blue}{0.1})^4\\&+\color{green}{252}\cdot 1^5 (\color{blue}{0.1})^5+\color{green}{210}\cdot 1^4 (\color{blue}{0.1})^6+\color{green}{120}\cdot 1^3\cdot (\color{blue}{0.1})^7+\color{green}{45}\cdot 1^2 (\color{blue}{0.1})^8\\&+\color{green}{10}\cdot 1 (\color{blue}{0.1})^9+(\color{blue}{0.1})^{10}\\\,\\\approx&2.5937424601\end{align}$$
Note: The digit $2$ present in the mathematical constant $e$ comes from:
We have: $\left(1+\frac{1}{n}\right)^n\,\,\text{(1)}$ and $1^n=1\quad\text{where $n\in\Bbb R$}$. That's why there will always be a $1$ left no matter how large $n$ is when we expand the expression $\text{(1)}$.
From the binomial theorem, the second coefficient in the expanded form of $(a+b)^n$ is just $n$. And therefore the second term in the expansion of $\left(1+\frac{1}{n}\right)^n$ will be: $n\times1^{n-1}\times\frac1n$ which is equal to $1$.
But in the limit the OP asks about, we are dividing $1$ by $\left(1+\frac{1}{n}\right)^n$, so it is natural that we get at the end $1/e$.
$$ \left(\dfrac{n}{n+1}\right)^n = \frac {1}{\left(\dfrac{n+1}{n}\right)^n}=\frac {1}{\left(1+\dfrac{1}{n}\right)^n}$$
Did you know that $\lim_{n\rightarrow \infty} (1+1/n)^n =e$? This you can use to find the limit.
I came at this problem a little differently than the other answers. We want to find $$L = \lim\limits_{n \to \infty} \left(\frac{n}{n+1}\right)^n.$$ Since the expression inside the limit is always positive, we should be able to take the log of the limit, then solve it (we just have to remember to transform back at the end). $$\begin{align} \log(L) = & \lim\limits_{n \to \infty} n \log \left( \frac{n}{n+1} \right) \\ = & \lim\limits_{n \to \infty} n \log \left( 1 - \frac{1}{n+1} \right). \end{align}$$ Since we're taking the limit of $\log(1 + x)$ as $x \to 0$, what we really care about is the first order behavior of $\log(1 + x)$, so we take the Taylor series: $$\log(1 + x) = x + O(x^2),$$ where $O(x^2)$ indicates higher-order terms (second-order and greater). So $\log(1 + x) \approx x$ for small $x$ ($x \approx 0$). Substituting back into our limit: $$\begin{align} \log(L) = & \lim\limits_{n \to \infty} n \left( -\frac{1}{n+1} \right) \\ = & -1. \end{align}$$ Therefore $L = e^{-1} = \frac{1}{e}$.
Hint:
$$\left(\frac{n}{n+1}\right)^n=\left(\frac{n+1-1}{n+1}\right)^n=\left(1-\frac{1}{n+1}\right)^n$$