# Limit of $\frac{1}{r}\ln\left(1+r\sum\limits_{i=1}^n p_i \ln(x_i)+ \omicron(r)\right)$

Let be $$n \in \mathbb{N}$$ arbitrary but fixed, $$\sum\limits_{i=1}^n p_i =1$$ and $$\forall ~ 1\leq i \leq n$$ we assume: $$x_i \in \mathbb{R}$$.

What is the limit of

$$\lim\limits_{r\to 0}~\frac{1}{r}\ln\left(1+r\sum\limits_{i=1}^n p_i \ln(x_i)+ \omicron(r)\right)$$, where $$\lim\limits_{r\to 0}\frac{\omicron(r)}{r}=0$$?

I tried some manipulations and the theorem of L'Hospital but it only got worse...

• This is not well stated. Is $n$ fixed? What are the $x_i?$ What are the $p_i?$ – zhw. Jun 2 at 22:21
• Thanks for the remark. I have edited the question. – Philipp Jun 2 at 22:27

Let $$S(r)=r\sum_{i=1}^np_i\ln(x_i)$$

we have $$\lim_{r\to 0}(S(r)+o(r))=0$$

thus

$$\color{red}{\lim_{r\to 0}\frac{\ln\Bigl(1+S(r)+o(r)\Bigr)}{S(r)+o(r)}=1}$$

therefore, the limit is $$\lim_{r\to0}\frac{S(r)+o(r)}{r}=\sum_{i=1}^np_i\ln(x_i)$$ because

$$\frac{\ln(1+S(r)+o(r))}{r}=$$ $$\color{red}{\frac{\ln(1+S(r)+o(r))}{S(r)+o(r)}}\frac{S(r)+o(r)}{r}.$$

and $$\frac{S(r)}{r}+\frac{o(r)}{r}=$$ $$\sum_{i=1}^np_i\ln(x_i)+\frac{o(r)}{r}$$

• Can you explain to me the step that follows "the limit is ..." a little bit more? Why can we conclude the limit of $\lim_{r\to0}\frac{S(r)+o(r)}{r}$ from $\lim_{r\to 0}\frac{\ln\Bigl(1+S(r)+o(r)\Bigr)}{S(r)+o(r)}=1$ ? – Philipp Jun 2 at 22:16
• @Philipp I just added some lines in the end. hope will help. – hamam_Abdallah Jun 2 at 22:32
• Thank you for the further explanations :) – Philipp Jun 2 at 23:49
• Do you know if it is possible to show the statement with the theorem of L'Hospital? The problem is that when applying this theorem there appears a term $\omicron'(r)$ and I am not sure what happens to this term when $r \to 0$. – Philipp Jun 3 at 0:01

Hint

Remembering that $$\lim_{r \to 0} \frac{\ln(1+r)}{r} = 1$$

we get $$\begin{split} \lim_{r \to 0}\frac{\ln\left(1+r\sum\limits_{i=1}^n p_i \ln(x_i)+ \omicron(r)\right)}{r} &= \lim_{r \to 0}\frac{\ln\left(1+r\sum\limits_{i=1}^n p_i \ln(x_i)+ \omicron(r)\right)}{\sum\limits_{i=1}^n p_i \ln(x_i)+ \omicron(r)}\cdot \frac{r\sum\limits_{i=1}^n p_i \ln(x_i)+ \omicron(r)}{r} \end{split}$$

Can you continue from here?