Section 10.7

Comparison Tests

Often we can't integrate or find partial sums. Instead, we compare our messy series to a nice, clean one we already know.

The Strategy of Guessing

Dominant Terms

For large , only the highest power matters. Ignore the constants and smaller powers to guess the behavior.

Example

Since

converges (p-series), we guess our series converges. Now we prove it.

Comparison Test

Also called the Direct Comparison Test. We use logic to "squeeze" the series.

Case 1: Convergence

If converges and , then converges.

"Smaller than finite is finite"

Case 2: Divergence

If diverges and , then diverges.

"Larger than infinite is infinite"

Warning: Useless Information

Being larger than a convergent series tells you nothing.
Being smaller than a divergent series tells you nothing.

Limit Comparison Test

This is often easier than Direct Comparison because you don't need to build a perfect inequality.

Often Preferred

The Condition

1If (finite and positive), then both series behave the same.
2Unlike DCT, you don't need to worry about "larger" or "smaller". Just pick a good comparison series!

Worked Examples

Example 1: Direct Comparison Test

Check .

1. Guess

Behaves like

, which Converges. We need to show our series is smaller.

2. Compare

(Denominator is larger)

(Fraction is smaller)

Since it is smaller than a convergent series, it Converges.

Example 2: Limit Comparison Test