Analytic Series

This exercise is from 'Basic Complex Analysis' by
Hoffman and Marsden (great title huh).

3.1.14. Prove that the series

g(z) = sum_{n = 1}^{oo} z^n / [1 + z^(2n)]

...converges in both the interior and exterior of the
unit circle and represents an analytic function in each
region.

In section 3.1 they basically introduced the Weierstrass
M-test and the analytic convergence theorem. I'm pretty
sure my proof for the interior is correct. However, I'm
getting stuck on the proof for the exterior. Here is
what I have so far:

PROOF. [Interior] Claim g(z) converges uniformly on the
disk D_r = {z : |z| <= r} for any r < 1. Now,

g_n(z) = z^n / [1 + z^(2n)]

...so...

|g_n(z)| <= r^n / [1 + r^(2n)]

...for any z in D_r. Let M_n = r^n / [1 + r^(2n)].

Then sum_{n = 1}^{oo} M_n converges since

lim_{n -> oo} | r^(n + 1) / [1 + r^(2n + 2)] * [1 + r^(2n)] / r^n |

= |r| lim_{n -> oo} | [1 + r^(2n)] / [1 + r^(2n + 2)] |

= |r| < 1.

Thus, the series g(z) converges uniformly on D_r via the
Weierstrass M-test. Moreover, each g_n(z) is analytic on
D_r and it follows from the analytic convergence theorem
that g(z) is analytic on D_r. []

I'm not sure how to get started on the exterior part.
Any suggestions/corrections would be appreciated.

Thanks in Advance,
Kyle
.

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