Question

Use the properties of geometric series to find the sum of the series. for what values of the variable does the series converge to this sum? 5+(−10)z+20z2+(−40)z3+⋯

Answer 1

$5\times(-2)^0=5$
$5\times(-2)^1=-10$
$5\times(-2)^2=20$
$5\times(-2)^3=-40$
...

There's enough of a pattern here to discern the series to be

$\displaystyle\sum_{k=0}^\infty 5(-2z)^k$

which converges as long as $|-2z|=2|z|$, or $|z|$. Under this condition, the series would converge to the function

$f(z)=\dfrac5{1+2z}$