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AP Calc BC Series Convergence Tests cheat sheet - grade 11-12

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Calculus Grade 11-12

AP Calc BC Series Convergence Tests Cheat Sheet

A printable reference covering geometric, p-series, comparison, ratio, root, alternating, and integral tests for grades 11-12.

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AP Calculus BC series convergence tests help students decide whether an infinite series has a finite sum. This cheat sheet organizes the major tests used on the AP exam so students can quickly choose an appropriate strategy. It is useful for homework, review, and checking the conditions that must be satisfied before using each test. The core ideas include recognizing special series, comparing terms to known benchmarks, and using limits to test long-term behavior. Important tools include the geometric series formula, the p-series rule, the nth-term test, the integral test, comparison tests, the alternating series test, and the ratio and root tests. Many problems require both choosing the correct test and clearly stating whether the series converges absolutely, converges conditionally, or diverges.

Key Facts

  • The nth-term test says if lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0 or the limit does not exist, then ∑n=1∞an\sum_{n=1}^{\infty} a_n diverges.
  • A geometric series ∑n=0∞arn\sum_{n=0}^{\infty} ar^n converges to a1−r\frac{a}{1-r} when ∣r∣<1|r|<1 and diverges when ∣r∣≥1|r| \geq 1.
  • A p-series ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p} converges when p>1p>1 and diverges when p≤1p \leq 1.
  • The integral test applies when f(x)f(x) is positive, continuous, and decreasing, and an=f(n)a_n=f(n), so ∑an\sum a_n and ∫1∞f(x) dx\int_{1}^{\infty} f(x)\,dx either both converge or both diverge.
  • The direct comparison test says if 0≤an≤bn0 \leq a_n \leq b_n and ∑bn\sum b_n converges, then ∑an\sum a_n converges.
  • The limit comparison test says if an>0a_n>0, bn>0b_n>0, and lim⁡n→∞anbn=L\lim_{n \to \infty} \frac{a_n}{b_n}=L where 0<L<∞0<L<\infty, then ∑an\sum a_n and ∑bn\sum b_n have the same behavior.
  • The ratio test uses L=lim⁡n→∞∣an+1an∣L=\lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|, where the series converges if L<1L<1, diverges if L>1L>1, and is inconclusive if L=1L=1.
  • The alternating series test says ∑(−1)nbn\sum (-1)^n b_n or ∑(−1)n+1bn\sum (-1)^{n+1}b_n converges if bn>0b_n>0, bn+1≤bnb_{n+1}\leq b_n, and lim⁡n→∞bn=0\lim_{n \to \infty} b_n=0.

Vocabulary

Infinite series
An infinite series is a sum of infinitely many terms, written as ∑n=1∞an\sum_{n=1}^{\infty} a_n.
Convergent series
A convergent series is an infinite series whose sequence of partial sums approaches a finite limit.
Divergent series
A divergent series is an infinite series whose partial sums do not approach a finite limit.
Absolute convergence
A series ∑an\sum a_n converges absolutely if ∑∣an∣\sum |a_n| converges.
Conditional convergence
A series ∑an\sum a_n converges conditionally if ∑an\sum a_n converges but ∑∣an∣\sum |a_n| diverges.
Partial sum
A partial sum is the finite sum SN=∑n=1NanS_N=\sum_{n=1}^{N} a_n, which is used to study the behavior of an infinite series.

Common Mistakes to Avoid

  • Using the nth-term test to prove convergence is wrong because the test can only prove divergence when lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0 or does not exist.
  • Applying the alternating series test without checking that bnb_n decreases is wrong because the test requires bn+1≤bnb_{n+1}\leq b_n and lim⁡n→∞bn=0\lim_{n \to \infty} b_n=0.
  • Forgetting absolute values in the ratio test is wrong because the ratio test uses ∣an+1an∣\left|\frac{a_{n+1}}{a_n}\right| to test absolute convergence.
  • Comparing a series to a larger divergent series is wrong because 0≤an≤bn0\leq a_n\leq b_n and divergent ∑bn\sum b_n does not tell you what happens to ∑an\sum a_n.
  • Treating an inconclusive result as a final answer is wrong because tests like the ratio test with L=1L=1 require choosing another convergence test.

Practice Questions

  1. 1 Determine whether ∑n=1∞3n5n\sum_{n=1}^{\infty} \frac{3^n}{5^n} converges or diverges, and find its sum if it converges.
  2. 2 Use an appropriate test to determine whether ∑n=2∞1n(ln⁡n)2\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2} converges or diverges.
  3. 3 Determine whether ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}} converges absolutely, converges conditionally, or diverges.
  4. 4 Explain why the ratio test is often a good choice for series involving factorials or powers, but may be inconclusive for many rational functions of nn.