I'm stuck with this limit $(1 - \frac{c}{n}\log n )^{1-n}$ as $n \rightarrow \infty$ where $c < 1$. I tried to plot the limit and it looks like it goes to infinity, although very slowly, but I can't prove it. Any ideas?
A Tricky Limit: $(1 - \frac{c}{n}\log n )^{1-n}$
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0Consider taking the limit of $(1-n)\log(1-(c\log n)/n)$... – 2011-10-10
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2[Wolfram Alpha](http://www.wolframalpha.com/input/?_=1318257578883&i=(1+-+%5cfrac%7bc%7d%7bn%7d%5clog+n+)%5e%7b1-n%7d+as+n+-%3e+infinity&fp=1&incTime=true) says, $\lim_{n\to\infty}(1-\frac cn\log n)^{1-n} = \infty$... – 2011-10-10
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1Hence the sequence is $n^{c+o(1)}$, which is slow but not so slow... – 2011-10-10
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1For $0
$\infty$ . – 2011-10-10 -
0I tried writting the expression as : $L:=\lim_{n \to \infty} \exp{(1-n+\frac{(n-c)}{n} \log n)}$. – 2011-10-10
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0Oops.. I messed up the expression by missing $\log$. Read $L:=\lim_{n \to \infty} \exp{(1-n) \log {(\frac{(n-c \log n)}{n}}}$ – 2011-10-10
3 Answers
Near $x=0$, $\log(1+x)=x+O(x^2)$ so as $n\to\infty$, we get that $$ \begin{align} (1-n)\log(1-\frac{c}{n}\log(n)) &=(1-n)\left(-\frac{c}{n}\log(n)+O\left(\left(\frac{\log(n)}{n}\right)^2\right)\right)\\ &=\frac{c(n-1)}{n}\log(n)+O\left(\frac{\log(n)}{n}\log(n)\right)\\ &\to\infty\text{ (like }c\log(n)\text{)} \end{align} $$ if $c>0$. Thus, $(1 - \frac{c}{n}\log n )^{1-n}\to\infty$ like $n^c$.
You can factor out $(1 - {c \over n}\log(n))$ which converges to $1$ and thus you are looking for $$\lim_{n \rightarrow \infty} (1 - {c \over n}\log(n))^{-n}$$ $$= \lim_{n \rightarrow \infty} \bigg((1 - {c \over n}\log(n))^{{n \over c \log n}}\bigg)^{-c\log n}$$ Since ${\displaystyle \lim_{\epsilon \rightarrow 0} (1 - r)^{1 \over r} = {1 \over e}}$, the expression inside the large parentheses goes to ${1 \over e}$ as $n$ goes to infinity. Since $({1 \over e})^{-c\log n} = n^c$, this means the expression diverges to infinity as $n^c$ does. (Well faster than $n^{c'}$ for any $c' < c$.)
Some graph samples shows that as x gets larger the limit goes larger.
Edit: I removed my steps due to corrections suggested below.
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0*As n grows larger and larger, (c/n)log(n) goes to c*... Hmmmm, well, it does not. – 2011-10-10
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0Well, it tends to $0$. And for $c \in (0,1)$, one might say that this limit tends to c. – 2011-10-10
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0Thanks to @DidierPiau and gaurav for help – 2011-10-10
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0@gaurav, ??? $ $ – 2011-10-10
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0I tried to address the comment to 2 people but I was not allowed to by the forum. – 2011-10-10