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Series & Convergence Tests cheat sheet - grade 11-12

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Calculus Grade 11-12

Series & Convergence Tests Cheat Sheet

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

TopicsCalculusIntegralsFunctionsAlgebraStudy Guide
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This cheat sheet covers infinite sequences, infinite series, and the main tests used to decide convergence. Students need it because series problems often depend on choosing the right test quickly and checking every condition. It organizes common tests, key formulas, and decision rules into a compact reference for grades 11-12 calculus.

The core idea is that a series ∑n=1∞an\sum_{n=1}^{\infty} a_n converges when its partial sums approach a finite limit. Some series have exact formulas, such as geometric series, while others require tests such as comparison, integral, ratio, root, or alternating series tests. The most important first step is always to check whether lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0, because a nonzero limit means the series diverges.

Key Facts

  • The series ∑n=1∞an\sum_{n=1}^{\infty} a_n converges if the sequence of partial sums SN=∑n=1NanS_N = \sum_{n=1}^{N} a_n approaches a finite limit as N→∞N \to \infty.
  • The nth-term test says that 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(n)=anf(n)=a_n is positive, continuous, and decreasing, and ∑n=1∞an\sum_{n=1}^{\infty} a_n converges exactly when ∫1∞f(x) dx\int_{1}^{\infty} f(x)\,dx converges.
  • The direct comparison test says that if 0≤an≤bn0 \leq a_n \leq b_n and ∑bn\sum b_n converges, then ∑an\sum a_n converges, while if 0≤bn≤an0 \leq b_n \leq a_n and ∑bn\sum b_n diverges, then ∑an\sum a_n diverges.
  • 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 that ∑n=1∞(−1)n+1bn\sum_{n=1}^{\infty} (-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

Sequence
A sequence is an ordered list of terms, usually written as {an}\{a_n\}, where each term depends on the index nn.
Series
A series is the sum of the terms of a sequence, usually written as ∑n=1∞an\sum_{n=1}^{\infty} a_n for an infinite series.
Partial Sum
A partial sum is the finite sum SN=∑n=1NanS_N = \sum_{n=1}^{N} a_n used to study the behavior of an infinite series.
Convergence
Convergence means that the partial sums SNS_N approach a finite number as N→∞N \to \infty.
Divergence
Divergence means that a series does not approach a finite sum, often because SNS_N grows without bound or fails to settle.
Absolute Convergence
A series ∑an\sum a_n is absolutely convergent if ∑∣an∣\sum |a_n| converges.

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.
  • Forgetting the conditions of the integral test is wrong because f(x)f(x) must be positive, continuous, and decreasing on the interval being tested.
  • Applying the ratio test and concluding divergence when L=1L = 1 is wrong because L=1L = 1 means the test is inconclusive, not that the series diverges.
  • Comparing in the wrong direction is wrong because an≤bna_n \leq b_n with a convergent larger series proves convergence, while bn≤anb_n \leq a_n with a divergent smaller series proves divergence.
  • Assuming every alternating series converges is wrong because the alternating series test also requires bnb_n to decrease and lim⁡n→∞bn=0\lim_{n \to \infty} b_n = 0.

Practice Questions

  1. 1 Determine whether ∑n=1∞3n2\sum_{n=1}^{\infty} \frac{3}{n^2} converges or diverges, and name the test used.
  2. 2 Use the ratio test to decide whether ∑n=1∞n!5n\sum_{n=1}^{\infty} \frac{n!}{5^n} converges or diverges.
  3. 3 Determine whether the alternating series ∑n=1∞(−1)n+11n\sum_{n=1}^{\infty} (-1)^{n+1}\frac{1}{n} converges absolutely, converges conditionally, or diverges.
  4. 4 Explain why checking lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0 is necessary before choosing a convergence test, but is not enough to prove convergence.

Understanding Series & Convergence Tests

The useful skill is not memorizing a long list of tests. It is noticing the shape of the term. Powers of n often point toward a p-series or a comparison with one.

Expressions containing a constant raised to n often behave like geometric series. Factorials usually point toward the ratio test because most factorial factors cancel when consecutive terms are divided. Roots raised to the nth power can point toward the root test.

For a fraction made from polynomials, compare the highest powers in the numerator and denominator. A term that behaves like one over n to the p power has the same long-term behavior as the matching p-series.

This idea is called asymptotic thinking. It focuses on what dominates when n becomes very large.

The zero-term check is only a one-way filter. If the terms fail to approach zero, the problem ends because the running total cannot settle. If the terms do approach zero, much more work is needed.

For example, one over n approaches zero, yet adding all those small values still grows without bound. The important issue is the rate of decrease. Terms like one over n shrink too slowly.

Terms like one over n squared shrink fast enough. This difference can seem minor in a table of values, but after infinitely many additions it determines the result. Students should avoid treating small terms as negligible just because they eventually become tiny.

Comparison tests work best when every term is nonnegative and a familiar benchmark is nearby. Direct comparison requires an inequality that holds for all sufficiently large values of n. Limit comparison is often easier when fractions have complicated algebra.

It compares the given term with a simpler positive term by examining their ratio at large n. A positive finite limiting ratio means the two series share the same convergence behavior. The integral test gives a geometric picture.

Each term can be viewed as the area of a rectangle under a decreasing curve. The rectangles and the improper integral may differ by some area, yet they either both produce a finite total or both keep growing. This connects series to accumulated quantities such as total distance from changing speeds, total energy in repeated signals, and error estimates in numerical calculations.

Alternating series need special care because cancellation can make a series converge even when the series of positive magnitudes diverges. This is conditional convergence. Absolute convergence is stronger.

It occurs when the series formed by making every term positive converges. The ratio and root tests usually establish absolute convergence, especially for power series, exponentials, and factorial expressions. For an alternating series that meets its conditions, the error after stopping at a certain term is no larger than the magnitude of the next term.

That fact makes infinite sums practical for approximating numbers and functions. When solving problems, write down why each test applies. Check positivity and decreasing behavior for the integral or alternating tests.

Check whether a ratio-test result equals one, since that result gives no decision. Careful condition checks prevent most series mistakes.