1. Radius and Interval of Convergence

2. Use the Ratio Test
Simple enough, right?
Now let’s complicate things a bit…

3. Use the Ratio Test
But wait! When x = 2 or x = 4, then the Ratio Test gives us a limit of 1
And a limit of 1 with the Ratio Test is inconclusive

4. The only solution is to… test each endpoint
Use the Ratio Test
The only solution is to…
test each endpoint
Which gives us the Harmonic Series
x = 4
…which diverges
Which gives us the Alternating Harmonic Series
x = 2
…which converges
So the true interval of convergence is

5. The Radius of Convergence is always half the size of the interval of convergence regardless of the endpoints
Radius of Convergence = 1
Radius of Convergence = 4