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Let $ z $ be a complex number We de ne jzj2 = n(z) = p zz = x2 + y2. I want to find the radius of convergence of $$ \sum_ {n=0}^ {\infty}z^ {n} $$ my intuition is that this series converges for $ z\in d\left (0,1\right) $ (open unit disk).
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Series and sequences throughout these notes we’ll keep running into taylor series and fourier se ries
It’s important to understand what is meant by convergence of series be fore getting to numerical analysis proper And dahlquist and bjorck, numerical methods. A complex series is a formal in ̄nite sum of complex numbers The reason is that complex numbers add another dimension to the number system, thus opening up other routes to the solution
Finally complex numbers lead to remarkable insights For example complex numbers show that there is a very close connection between sinusoidal oscillations and exponential decay in a physical system. The complex numbers are formally de ned as the eld c = r[i], where i2 = 1 They are represented in the euclidean plane by z = (x
Y) = x + iy
The number i is the one with positive imaginary part
