Power series representations for geometric series
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Power series representations for geometric series

[From: ] [author: ] [Date: 11-11-28] [Hit: ]
............
Let z be a complex number
f(z) = -1 / (1 - z)
I understand that this can be represented by a power series in a disc D(0, 1)

But how can I represent it as a power series in a disc centered at 3 + 4i?

Everywhere I look on the internet, it only tells me how to solve this when z0 = 0... Thanks!

-
f(z) = -1/(1 - z)
......= -1/(1 - (z - (3 + 4i)) - (3 + 4i)), adding 0 cunningly
......= -1/[(-2 - 4i) - (z - (3 + 4i))]
......= 1/[(2 + 4i) (1 - -(z - (3 + 4i))/(2 + 4i)]
......= [1/(2 + 4i)] * Σ(n = 0 to ∞) [-(z - (3 + 4i))/(2 + 4i)]^n, via geometric series
......= Σ(n = 0 to ∞) (-1)^n (z - (3 + 4i))^n / (2 + 4i)^(n+1).

I hope this helps!
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keywords: geometric,series,for,representations,Power,Power series representations for geometric series
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