The Direct Comparison Test is a tool for deciding whether an infinite series converges or diverges by comparing it to a series you already understand. It is especially useful when the series has positive terms and looks similar to a p-series, geometric series, or harmonic series. Instead of finding an exact sum, you prove that the unknown series is smaller than a convergent benchmark or larger than a divergent benchmark.
This makes comparison one of the most practical tests in early calculus.
Key Facts
- Direct Comparison Test for convergence: if 0 <= a_n <= b_n for all n >= N and sum b_n converges, then sum a_n converges.
- Direct Comparison Test for divergence: if 0 <= b_n <= a_n for all n >= N and sum b_n diverges, then sum a_n diverges.
- The test requires nonnegative terms, usually a_n >= 0 and b_n >= 0 for all sufficiently large n.
- A p-series sum 1/n^p converges if p > 1 and diverges if p <= 1.
- A geometric series sum ar^n converges if |r| < 1 and diverges if |r| >= 1.
- Only the tail matters: changing or ignoring finitely many terms does not affect convergence or divergence.
Vocabulary
- Infinite series
- An infinite series is the sum of infinitely many terms, written as sum a_n.
- Convergence
- A series converges if its sequence of partial sums approaches a finite number.
- Divergence
- A series diverges if its partial sums do not approach a finite number.
- Benchmark series
- A benchmark series is a familiar series, such as a p-series or geometric series, used for comparison.
- Bounding series
- A bounding series is a series placed above or below another series using inequalities.
Common Mistakes to Avoid
- Comparing in the wrong direction for convergence. To prove convergence, the unknown series must be less than or equal to a convergent benchmark, not greater than it.
- Comparing in the wrong direction for divergence. To prove divergence, the unknown series must be greater than or equal to a divergent benchmark, not less than it.
- Using the test with negative or sign-changing terms without justification. The Direct Comparison Test applies to nonnegative terms, so absolute values or another test may be needed.
- Forgetting that inequalities only need to hold eventually. A few early terms can fail the comparison, because finitely many terms do not change whether a series converges.
Practice Questions
- 1 Use the Direct Comparison Test to determine whether sum from n = 1 to infinity of 1/(n^2 + 5) converges or diverges.
- 2 Use the Direct Comparison Test to determine whether sum from n = 2 to infinity of 1/(3n - 1) converges or diverges.
- 3 A student wants to prove that sum a_n converges by showing a_n >= 1/n^2 for all n. Explain why this comparison does not prove convergence, and state what inequality would be useful instead.