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Suppose we know a function f(z) is analytic in the finite z plane apart from singularities at z = i and z=-1. Moreover, let f(z) be given by the Taylor series:

[tex]f(z)=\displaystyle\sum_{j=0}^{\infty}a_{j}z^{j}[/tex]

where aj is known. Suppose we calculate f(z) and its derivatives at z = 3/4 and compute a Taylor series in the form

[tex]f(z)=\displaystyle\sum_{j=0}^{\infty}b_{j}\left(z-\frac{3}{4}\right)^{j}[/tex]

1. Where would this series converge?

2. How could we use this to compute f(z)?

3. Suppose we wish to compute f(2.5); how could we do this by series method?

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Do they want the radius of convergence here? Or do they want some general statement about the function it would converge to, that is some kind of expression in z using aj?

Since we know the value of the function at all of the derivations when z=.75 I said that

f(.75) = a, f'(.75) = b, f''(.75) = c ... so now I have a sequence of constants: a, b, c, d...

But, I don't know how to proceed because I'm unclear on what they are asking. When they say "Where would this series converge?"