Calculus BC Convergence Tests Summary
In AP Calculus BC, determining whether an infinite series converges or diverges is a fundamental skill. An infinite series is the sum of the terms of an infinite sequence. If the limit of the partial sums exists and is finite, the series converges; otherwise, it diverges. Below is a summary of the primary tests used to determine convergence.
1. The n-th Term Test for Divergence
Statement: If lim (n) an 0, then the series an diverges.
Important Note: If the limit is 0, the test is inconclusive. The series might converge or it might diverge. This test can only prove divergence, never convergence.
Best Used For: A quick check to rule out convergence.
2. Geometric Series
Form: arn-1 (starting at n=1) or arn (starting at n=0).
Condition: The series converges if and only if |r| < 1.
Sum Formula: If convergent, the sum is S = a / (1 - r).
Best Used For: Series with a constant ratio between successive terms.
3. p-Series
Form: (1/np).
Condition: The series converges if p > 1 and diverges if p 1.
Special Case: When p = 1, it is the Harmonic Series, which diverges.
Best Used For: Comparing other series (Comparison Tests).
4. Integral Test
Conditions: Let f(n) = an. The function f must be continuous, positive, and decreasing for x N.
Statement: The series an and the improper integral f(x) dx (from N to ) either both converge or both diverge.
Best Used For: Series where the term an looks like a function easy to integrate (e.g., 1/(n ln n)).
5. Direct Comparison Test (DCT)
Conditions: Assume 0 an bn for all n.
- If bn converges, then an converges.
- If an diverges, then bn diverges.
Best Used For: Rational functions or series resembling p-series/geometric series where terms can be clearly bounded.
6. Limit Comparison Test (LCT)
Conditions: Let an and bn be positive series. Compute L = lim (n) (an / bn).
- If 0 < L < , then both series either converge or diverge together.
- If L = 0 and bn converges, then an converges.
- If L = and bn diverges, then an diverges.
Best Used For: Similar to DCT, but used when the inequality an bn is hard to prove directly. Often compares to a p-series.
7. Ratio Test
Procedure: Compute L = lim (n) |an+1 / an|.
- If L < 1, the series is Absolutely Convergent (Converges).
- If L > 1 (or ), the series Diverges.
- If L = 1, the test is Inconclusive.
Best Used For: Series involving factorials (n!), exponentials (kn), or products.
8. Root Test
Procedure: Compute L = lim (n) n|an|.
- If L < 1, the series is Absolutely Convergent (Converges).
- If L > 1 (or ), the series Diverges.
- If L = 1, the test is Inconclusive.
Best Used For: Series where the term an is raised to the n-th power.
9. Alternating Series Test (Leibniz Test)
Form: (-1)n-1 bn or (-1)n bn where bn > 0.
Conditions: The series converges if both conditions are met:
- lim (n) bn = 0
- bn+1 bn for all n (the sequence of terms is decreasing).
Types of Convergence
Understanding the type of convergence is crucial, especially for alternating series.
| Type | Definition | Implication |
|---|---|---|
| Absolute Convergence | |an| converges. | The series an converges. |
| Conditional Convergence | an converges, but |an| diverges. | Convergence relies on the cancellation of signs. |
Quick Decision Strategy
When facing a series, follow this general order of operations to select a test:
- Is it geometric? Check ratio r.
- Is it a p-series? Check exponent p.
- n-th Term Test: Does the limit go to 0? If not, it diverges immediately.
- Is it alternating? Try the Alternating Series Test.
- Are factorials or exponentials involved? Use the Ratio Test.
- Are terms raised to the n power? Use the Root Test.
- Is it a rational function (polynomial/polyn)? Compare to a p-series using Limit Comparison or Direct Comparison.
- Can the term be easily integrated? Try the Integral Test.
Convergence/divergence Of Non Negative Term Series and Reference File Download Link
