Convergence of complex power series $\sum_n \dfrac{(-4+4\sqrt3i)^n}{n} \dfrac{1}{(z+3)^{6n}}$

Question

The problem

Find where does the following series converge and where does it converge absolutely $\sum_n \dfrac{(-4+4\sqrt3i)^n}{n} \dfrac{1}{(z+3)^{6n}}$.

My attempt. I first took the following substitution $w=\dfrac{1}{(z+3)^6}$, so I'm left with the following power series $\sum_n \dfrac{(-4+4\sqrt3i)^n}{n} w^n $ which I can find the radius of convegernce in terms of $|w|$ by using the Cauchy-Hadamard formula. Now I'm left to see if there's convergence on the border of that region. For that I'm having trouble deciding. I can't use Dirichlet criterion because the main sequence is not decreasing nor I can use Dedekind criterion. So I'm led to believe that maybe the series doesn't converge on the border, but I can't find a suitable series to compare to.

For what it's worth, if I made the problem a little bit easier, for example by taking $w=\dfrac{1}{(z+3)}$ I can find the radius of convergence of $\sum_n w^n$, but I can't relate that with $\sum_n a_n w^n$ if $a_n$ doesn't converge to 0.

Any hints would be apreciated.

Answer 1

First note that $-4+i4\sqrt 3=8e^{i2\pi/3}$. Hence, we see that

$$\begin{align}\left|\frac{\left(-4+i4\sqrt 3\right)^n}{(z+3)^{6n}}\right|&=\left|\frac{8e^{i2\pi/3}}{(z+3)^6}\right|^n\\\\ &=\left(\frac{8}{|z+3|^6}\right)^n \end{align}$$

Finally, taking the $n$'th root reveal

$$\begin{align} \lim_{n\to\infty}\sqrt[n]{\left|\frac{\left(-4+i4\sqrt 3\right)^n}{n(z+3)^{6n}}\right|}&=\frac{8}{|z+3|^6}\lim_{n\to\infty}n^{-1/n}\\\\ &=\frac{8}{|z+3|^6}<1 \end{align}$$

when $|z+3|>8^{1/6}$. So, the series converges for $|z+3|>8^{1/6}$.

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What Happens on the Circle $|z+3|=8^{1/6}$? Well, if $(z+3)=8^{1/6}e^{i\theta}$, then

$$\begin{align} \sum_{n=1}^\infty \frac{(-4+i4\sqrt3)}{n(z+3)^{6n}}&=\sum_{n=1}^\infty \frac{e^{i2n\pi/3}}{ne^{i6n\theta}}\\\\ &=\sum_{n=1}^\infty \frac{e^{i(6\theta +2\pi/3)n}}{n} \end{align}$$

Now, apply Dirichlet's Test to find that range of values of $\theta$ for which there exists a number $N$ such that

$$\left|\sum_{n=1}^N e^{i(6\theta+2\pi/3)n}\right|<M$$

for all $N$.

Can you finish now?

Answer 2

Note that$$\sqrt[n]{\left\lvert\dfrac{(-4+4\sqrt3i)^n}{n} \dfrac{1}{(z+3)^{6n}}\right\rvert}=\frac{\left\lvert-4+4\sqrt3i\right\rvert}{\sqrt[n]n\lvert z+3\rvert^6}=\frac8{\sqrt[n]n\lvert z+3\rvert^6}\to\frac8{\lvert z+3\rvert^6}.$$Therefore, the series converges if $\lvert z+6\rvert>\sqrt2$ and diverges if $\lvert z+3\rvert<\sqrt2$.

If $\lvert z+3\rvert=\sqrt2$, there are two possibilities:

• $(z+3)^6=-4+4\sqrt3i$: then your series is the harmonic series, which diverges.
• $\lvert z+3\rvert=\sqrt2$, but $(z+3)^6\neq-4+4\sqrt3i$: then your series converges, by Dirichlet's test.