Is There a Power Series That Converges at One Point and Diverges at Another?

mariab89
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Homework Statement



does there exist a power series that converges at z= 2+31 and diverges at z=3-i

Im really stuck on this one! any ideas?
 
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Do you mean 2+3i and 3-i? And do you mean a power series centered at z=0? There is a theorem about convergence of power series based on a radius of convergence. Can you find it?
 
Yes, sorry my question is to determine whether there exists a power series that converges at z = 2 + 3i and diverges at z = 3 - i.
 
Ok, so is the power series just a sum of z^i (as opposed to (z-c)^i)? And what do you know about 'radius of convergence'?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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