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Convergence or Not on the Boundary of the Disc of Copnvergence

Every power serieshas a disc of convergence. The radius of the disc may be 0, infinity, or some finite number.

If the radius of convergence is some finite number r, then the power series converges for all z<r and diverges for all z>r, but for z=r, the series may either converge or diverge, or converge at some points absz}=r and diverge for others.

converges for alland diverges for allIfthe series also diverges.

has radius of convergence 1 and converges at every point of the disc of convergenceby comparison with the serieswhich is convergent.

has radius of convergence 1.

If z=1 the series becomes which diverges.

If z=-1 the series becomes which converges so the series converges at every point of the disc of convergenceby comparison with the serieswhich is convergent.