A "spiral" in the Complex plane

[Hill]

Homework Statement

Starting from the origin, go one unit east, then the same length north, then (1/2) of the previous length west, then (1/3) of the previous length south, then (1/4) of the previous length east, and so on. What point does this “spiral” converge to?

Relevant Equations

series sum

I understand that the "spiral" converges to 1+i-1/2-i/3!+1/4!+i/5!-1/6!-i/7!... .
It splits into two: one for Re, 1-1/2+1/4!-1/6!..., and the other for Im, 1-1/3!+1/5!-1/7!... .
Any hints on how to compute them?

 

[pasmith]

What are $\displaystyle\sum_{n=0}^\infty \dfrac{(-1)^n x^{2n}}{(2n)!}$ and $\displaystyle\sum_{n=0}^\infty \dfrac{(-1)^n x^{2n+1}}{(2n+1)!}$ when $x = 1$?

 

[Hill]

What are $\displaystyle\sum_{n=0}^\infty \dfrac{(-1)^n x^{2n}}{(2n)!}$ and $\displaystyle\sum_{n=0}^\infty \dfrac{(-1)^n x^{2n+1}}{(2n+1)!}$ when $x = 1$?

$\cos$ and $\sin$, of course. Thanks!

 

[Orodruin]

Or, even better:
$$
\sum_{n=0}^\infty \frac{(ix)^n}{n!}
$$
…

 

[Hill]

Or, even better:
$$
\sum_{n=0}^\infty \frac{(ix)^n}{n!}
$$
…

Yes. Straight.