Find the function f(z) = 1/(1-z) given in a Taylor series around i. Specify the maximum disk where this representation is valid
Thanks for your help
Thanks for your help
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1/(1 - z)
= 1/((1 - i) - (z - i))
= 1/[(1 - i)(1 - (z - i)/(1 - i))]
= (1/(1 - i)) * Σ(n = 0 to ∞) ((z - i)/(1 - i))^n, via geometric series
= Σ(n = 0 to ∞) (z - i)^n / (1 - i)^(n+1).
This converges when |(z - i)/(1 - i)| < 1 <==> |z - 1| < √2.
So, the maximal radius R = √2.
I hope this helps!
= 1/((1 - i) - (z - i))
= 1/[(1 - i)(1 - (z - i)/(1 - i))]
= (1/(1 - i)) * Σ(n = 0 to ∞) ((z - i)/(1 - i))^n, via geometric series
= Σ(n = 0 to ∞) (z - i)^n / (1 - i)^(n+1).
This converges when |(z - i)/(1 - i)| < 1 <==> |z - 1| < √2.
So, the maximal radius R = √2.
I hope this helps!