Recipe

Test the endpoints and complete the interval of convergence

Use when: the radius R and open interval (c-R,c+R) have already been found (recipe above), and the two endpoints still need to be resolved

Goal: the complete interval of convergence, with each endpoint correctly included or excluded

Taught in Testing the endpoints · extends Find the radius of convergence and the open interval

1. 1

   Plug x=c+R into the power series; this eliminates x entirely, leaving an ordinary series in n alone.

   Example

   Continuing \displaystyle\sum_{n=1}^\infty\frac{(x-2)^n}{n\cdot3^n}, with R=3, c=2: at x=5, (x-2)^n=3^n, giving \displaystyle\sum_{n=1}^\infty\frac{3^n}{n\cdot3^n}=\sum_{n=1}^\infty\frac1n, the harmonic series.

   Why: substituting a specific endpoint value collapses the power series to exactly the kind of numerical series every earlier convergence test in this course was built for.

2. 2

   Reach for the convergence test that matches the resulting series' structure (Alternating Series Test, p-series, comparison, ratio test on the numbers, and so on).

   Example

   \displaystyle\sum\frac1n is the harmonic series, a p-series with p=1, which diverges.

   Why: the Ratio Test already ruled out this endpoint as automatic; a completely different tool, chosen for what this specific numerical series looks like, is what actually settles it.

3. 3

   Substitute x=c-R into the power series and apply whichever test fits that resulting series, entirely independently of the right endpoint's result.

   Example

   At x=-1: (x-2)^n=(-3)^n, giving \displaystyle\sum_{n=1}^\infty\frac{(-3)^n}{n\cdot3^n}=\sum_{n=1}^\infty\frac{(-1)^n}{n}, the alternating harmonic series, which converges (conditionally) by the Alternating Series Test.

   • a series converges at one endpoint but diverges at the other — the endpoint is included where it converges and excluded where it diverges.
   • a series converges (or diverges) at both endpoints — both endpoints are included, or both excluded, together.

   Why: there's no shortcut that infers one endpoint's behavior from the other's; a series can converge at one edge and diverge at the other, so each has to be tested on its own.

4. 4

   Combine the open interval from the earlier recipe with the correct bracket or parenthesis at each endpoint, based on Steps 2 and 3.

   Example

   x=-1 converges (include it); x=5 diverges (exclude it): \boxed{\text{Interval of convergence: } [-1,5)}

   Why: this is the complete, working answer to "for which x does this power series converge," the question the radius alone only partially answered.