Determine the radius of convergence of the power series:

Test for convergence at the boundary points if is finite.

We have

Then applying the ratio test we have

Hence, the radius of convergence is 1, and the series converges for .

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Stumbling Robot

A Fraction of a Dot
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Determine the radius of convergence of *∑ ((1*3*5*…*(2n-1)) / (2*4*6*…*(2n)))*^{3} z

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Determine the radius of convergence of the power series:

Test for convergence at the boundary points if is finite.

We have

Then applying the ratio test we have

Hence, the radius of convergence is 1, and the series converges for .

Should it also converge at boundary since the real terms are proven to converge by Gauss test somewhere in the book?

yes, it also coverges at |z|=1, z1, using Dirichlet’s test

z not equal to 1