Sign in to save

Bookmark this page so you can find it later.

Sign in to save

Bookmark this page so you can find it later.

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

Click image to open full size

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.

TopicsCalculusIntegralsFunctionsAlgebraStudy Guide
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

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.

Understanding AP Calc BC Series Convergence Tests

A series is understood through its partial sums. Start by adding a finite number of terms, then watch what happens as more terms are included. If these running totals settle near one fixed number, the series converges.

If they grow without bound, jump around, or approach different values, it diverges. This viewpoint explains why the individual terms must approach zero. Terms that remain noticeably large keep changing the running total.

But terms approaching zero do not guarantee convergence. The harmonic series is the classic warning. Its terms shrink forever, yet their accumulated total still grows.

The best test usually comes from the shape of the general term. Exponentials and powers with a fixed common multiplier often point to a geometric series. Expressions involving factorials, products of consecutive integers, or a number raised to the nth power often simplify well under the ratio test.

Terms raised to the nth power can be good candidates for the root test. Fractions built from polynomials usually behave like a power of n. Focus on the highest powers because lower-degree terms become less important for large n.

For comparison, ignore constants when they only scale every term by a fixed positive amount. A series with terms similar to one over n squared has the same long-term behavior as that familiar benchmark.

Positive-term tests work by measuring size. Direct comparison requires an inequality in the useful direction, which is a common source of mistakes. To prove convergence, place the target terms below terms from a known convergent series.

To prove divergence, place the target terms above terms from a known divergent series. Limit comparison is often faster when expressions are complicated but have matching dominant terms. The integral test connects a sum to areas under a curve.

Each term can be pictured as the height of a rectangle. For a decreasing positive function, the rectangles and the area differ by only a controlled amount, so they share the same final behavior. This connection matters in physics when sums approximate accumulated quantities such as distance, energy, or probability.

Alternating series need extra care because positive and negative terms can cancel. A series may converge conditionally when its signed terms converge but the series made from their absolute values diverges. Absolute convergence is stronger because it guarantees convergence even if the terms are rearranged.

Conditional convergence is more delicate. On exam work, state the test conditions before giving a conclusion, especially decrease and the limit of the positive magnitudes for an alternating series. The ratio and root tests have an important blind spot when their limit equals one.

That result says nothing, so switch to another method rather than claiming divergence or convergence. A reliable routine is to check the term limit first, identify the term pattern, verify every condition, then name the conclusion precisely.