Page 101 - Calculus for the Clueless, Calc II

Let's look at .

This is a geometric series that converges for

Thus, the region of convergence is |x - 4| < 3 or 1 < x < 7. Test x = 1 and substitute into the original series. We
get

which diverges (Example 11). For x = 7, we get

which diverges (Example 10).

Example 36—

!, a nice one.

This says no matter what x is, the limit will always be less than 1. The region of convergence is all real
numbers.

Example 37—

except if x = 0. The region of convergence is just the point x = 0.

Which Test To Use

After finishing the original draft of this book but before running off copies, I finished student-testing this
section on infinite series. It became absolutely clear that this page is necessary.

1. Always see if the terms go to zero first. If they don't, the series diverges. If the terms go to zero, the series at
least converges Conditionally if it alternates.