Interval of Convergence

After that secondly, the interval of all x's, involving the endpoints if need be, for which the power series converges is termed as the interval of convergence of the series.

These two notions are quite closely tied together.  If we be familiar with that the radius of convergence of a power series is R after that we have the subsequent.

a- R < x < a + R                      power series converges

x < a - R and x > a + R          power series diverges

The interval of convergence must then consist of the interval a - R < x < a + R as we know that the power series will converge for these values.  We as well know that the interval of convergence can't consist of x's in the ranges x < a - R and x > a + R as we know the power series diverges for these value of x.  Hence, to completely identify the interval of convergence all that we have to do is find out if the power series will converge for  x = a - R or x = a + R.  If the power series converges for one or both of these values after that we'll need to involve those in the interval of convergence.