I have to solve these problems and show how to solve them in front of students tomorrow though I haven't learned the way to calculate infinite series of complex numbers.
(1) $\sum _{n=0} ^{\infty} z^n$
(2) $\sum _{n=0} ^{\infty} (3^{n+1} - 2^{n+2}) z^n$
Could you tell me the solutions or hints? Thank you in advance.
$\endgroup$ 21 Answer
$\begingroup$Let $S_z=\sum_{n=0}^\infty z^n:$
$$(1-z)S_z=(1-z)(1+z+z^2+\dots)\\=(1\color{red}{+z+z^2+z^3+\dots})-(\color{red}{z+z^2+z^3+\dots})\\=1$$
$$S_z=\frac1{1-z}$$
Which is only true if $|z|<1$ so the sum converges.
$$\sum_{n=0}^\infty(3^{n+1}-2^{n+2})z^n=3\sum_{n=0}^\infty(3z)^n-4\sum_{n=0}^\infty(2z)^n$$
Which is of the form
$$=3S_{3z}-4S_{2z}$$
And then use the above with the condition $|z|<1/3$
$\endgroup$ 3
