Power series convergence depends on the values of x input into the series. A power series may converge for some x values and not others. The set of points where the series converges is called the interval of convergence. If a power series is centered at x = a, then the power series always converges at its center. For example, a geometric series converges for x in the interval (-1,1) but diverges for x outside this interval.

2. Differential Equation

14.  power series convergence is still a major
question. The difference is that the
convergence of the series will now depend
upon the values of ”x” that we put into the
series. A power series may converge for some
values of ”x” and not for other values of x.
 The set of points where the series converges
is called the interval of convergence.

15.  If a power series centered at “x =a” then
power series always converges at its center.
For example a geometric series 𝑛=0
∞
𝑥 𝑛
converges for all “x” in the interval (-1,1) but
diverge for all “x” that lies outside the
interval.