# Limit of $-itc\sqrt{n}+n\log(1-\frac{it}{c\sqrt{n}})$

For $$c>0, t\in \mathbb{R}$$ why $$-itc\sqrt{n}+nc\sum_{k\geq 0}\frac{1}{k+1}\bigl(\frac{it}{\sqrt{n}}\bigr)^{k+1}\stackrel{n\rightarrow \infty}\longrightarrow \frac{-ct^2}{2}?$$ Just because it isn't very clear, the summation was $$\log(1-\frac{it}{\sqrt{n}})$$ where $$\log$$ is the complex logarithm.

I only see it converging to $$-\infty$$.

• Separate the first 2 terms of the sum and simplify. What do you get? Write it down here.
– Gary
Aug 23, 2021 at 14:37
• Since $\lim_{n\to\infty}\dfrac 1{n^{1+t}}=0$ for $t\ge 0$, all but the first two terms of the sum vanish asymptotically. Aug 23, 2021 at 14:46
• We have that the real part of the sequence $\frac{1}{2}\log(1+\frac{t^2}{c^2n})\rightarrow 0$. So, I don't see how a sequence of the form $\dot{\imath} b_n$ converges to a real number other than zero while $b_n$ is a real sequence.
– Medo
Aug 23, 2021 at 14:53
• There is also a missing $\frac{1}{k+1}$ in the sum and something is wrong with the placing of $c$. Could you check this and perhaps write the original problem with the $\log$ in it properly?
– Gary
Aug 23, 2021 at 15:33
• @Gary The original problem: if $c>0, (X_n)_{n\geq1}$sequence of random, independent and identically distributed variables such that $X_n$~Gamma(nc, n) $\forall n\geq1, S_n=X_1+...+X_n$ Show that $S_n/n \stackrel{distr}{\rightarrow}Z$ with Z~N(0,c). I used the characteristic functions of the variables. Aug 23, 2021 at 15:50

What I have written at first was wrong. That was causing me the problem. Gary's remark helped. Solution: Separate the first 2 terms of the sum and simplify. What you get is $$-itc\sqrt{n}+itc\sqrt{n}+\frac{(it)^2c}{2}$$ + something converging to 0.