# Is the limit not infinity?

Is the limit of this not infinity? No matter what the value of p is? Or is there a way to simplify that fraction?

$$\lim_{k \to \infty} 2^{p}\left(\frac{k}{k+1}\right)^k$$

• Is that $2^p or 2^k$? If it is p as it looks you can remove that as it is a constant. Apr 19, 2013 at 18:43
• Answer to title question: "it's not." Apr 19, 2013 at 18:58
• Actually, the answer "it's not" would imply that the term goes to $\infty$ ;-) Apr 19, 2013 at 22:36
• There is no such limit as infinity, which is not a number. A limit expression either converges on a value (a limit exists and is that value) or else there does not exist a limit: as the limit-generating parameter increases or decreases, the value oscillates without settling, or grows without bound.
– Kaz
Apr 19, 2013 at 22:52
• @Kaz: It's pretty common to say that if for any positive number $M$ there's an $n$ such that from then on the sequence stays above $M$, then the limit is infinity. Apr 20, 2013 at 0:13

Hint: $\displaystyle \lim \limits_{k\to +\infty} 2^p\left(\frac{k}{k+1}\right)^k=2^p\lim \limits_{k\to +\infty} \left(\frac{k}{k+1}\right)^k=2^p \left[\lim \limits_{k\to +\infty} \left(\frac{k+1}{k}\right)^k\right]^{-1}$

• The last equality is a bit dangerous if you don't know anything about the limit (just to make sure the thread starter knows that). Apr 19, 2013 at 22:41
• @andreas I agree, but since I meant to give a hint and not a full answer, I chose to answer this way. Apr 20, 2013 at 0:37

The limit involves the quantity $k$, while the term $2^p$ does not. So you can take that out of the limit

$$2^p \lim_{k\rightarrow\infty} \left(\frac{k}{k+1}\right)^k$$

That limit should begin to look familiar. In particular, we know that Euler's constant $e$, is given by:

$$e = \lim_{k\rightarrow\infty} \left(1+\frac{1}{k}\right)^k$$

You can work with your limit to try and get it in that form and see if an $e$ pops our somewhere. Hint, it does :-)

Moving the constant out, you only have to deal with $\lim\limits_{k\to +\infty}\left(\frac{k}{k+1}\right)^k$. This is a $1^\infty$ indeterminate form, which you can analyze by setting $y=\left(\frac{k}{k+1}\right)^k$ and then examining $\ln y=k\ln\left(\frac{k}{k+1}\right)=\frac{\ln\left(\frac{k}{k+1}\right)}{\frac{1}{k}}$.

Using L'Hopital you get that $\lim\limits_{k\to +\infty}\ln y=1$ and so $\lim\limits_{k\to +\infty} y=e$.

• But now seeing all of the other answers, it is definitely more efficient to shortcut with recognizing the limit for $e$ :) Apr 19, 2013 at 18:57
• Good edits, much better than not recognizing the $1^\infty$ problem Apr 19, 2013 at 19:32

Use $\frac{k}{k+1} = \frac{k+1-1}{k+1} = 1 - \frac{1}{k+1}$ and $e^x = \lim_{k \leftarrow \infty} (1+\frac{x}{k})^k$. Clearly you get

$$\lim_{k \rightarrow \infty} 2^p \left(\frac{k}{k+1}\right)^k = 2^p \lim_{k \rightarrow \infty}\left(1 - \frac{1}{k+1}\right)^k = 2^p \lim_{k \rightarrow \infty}\left(1 + \frac{-1}{k}\right)^{k-1} = 2^p e^{-1} = 2^p/e$$

You can notice that

$$(\frac{k}{k+1})^k = \exp(k\ln(\frac{k}{k+1})) = \exp(k\ln(1-\frac{1}{k+1}))$$

Now if you remember that $\lim_{x\to -\infty} ln(1+\frac{1}{x}) \sim \frac{1}{x}$...

• Nice approach, but there should be a - within your log Apr 19, 2013 at 22:34
• You're right ! Thanks, I edited it. Apr 19, 2013 at 22:44