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.

Convergence Tests Decision Flowchart with Examples cheat sheet - grade college

Click image to open full size

Calculus Grade college

Convergence Tests Decision Flowchart with Examples Cheat Sheet

A printable reference covering convergence test flowcharts, geometric, p-series, comparison, limit comparison, ratio, root, integral, alternating, and divergence tests for college.

TopicsFunctionsGraphsPatternsExponentsIntegralsCalculusStudy Guide
Download PNG

Related Tools

Related Labs

Related Worksheets

Related Infographics

Study as Flashcards

This cheat sheet helps students choose the right convergence test for infinite series quickly and confidently. It organizes the main tests into a decision flowchart, then connects each decision to a formula and a typical example. College calculus students need this reference because many series look similar, but small structural differences determine which test works best.

The goal is to make test selection systematic instead of based on guessing.

The most important first step is checking the term test by evaluating lim⁡n→∞an\lim_{n \to \infty} a_n. From there, recognizable forms such as geometric series ∑arn\sum ar^n, p-series ∑1np\sum \frac{1}{n^p}, alternating series, factorials, exponentials, and positive rational expressions suggest specific tests. Comparison, limit comparison, ratio, root, integral, and alternating series tests each answer different kinds of convergence questions.

A polished flowchart should guide students from the form of ana_n to the most efficient test and then to a clear conclusion.

Key Facts

  • The divergence test says if lim⁡n→∞an≠0\lim_{n \to \infty} a_n \ne 0 or the limit does not exist, then ∑an\sum a_n diverges.
  • A geometric series ∑n=0∞arn\sum_{n=0}^{\infty} ar^n converges when ∣r∣<1|r| < 1 and has sum a1−r\frac{a}{1-r}.
  • A p-series ∑n=1∞1np\sum_{n=1}^{\infty} \frac{1}{n^p} converges when p>1p > 1 and diverges when p≤1p \le 1.
  • The integral test applies when an=f(n)a_n = f(n) and f(x)f(x) is positive, continuous, and decreasing, and ∑an\sum a_n converges exactly when ∫1∞f(x) dx\int_1^{\infty} f(x)\,dx converges.
  • The direct comparison test says if 0≤an≤bn0 \le a_n \le 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=c\lim_{n \to \infty} \frac{a_n}{b_n} = c with 0<c<∞0 < c < \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|: 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 converges if bn≥0b_n \ge 0, bn+1≤bnb_{n+1} \le 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.
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.
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
The NNth partial sum is SN=∑n=1NanS_N = \sum_{n=1}^{N} a_n, the sum of the first NN terms.

Common Mistakes to Avoid

  • Using the divergence test to prove convergence is wrong because lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0 is necessary but not sufficient for convergence.
  • Applying the ratio test when L=1L = 1 and claiming convergence or divergence is wrong because the ratio test is inconclusive at L=1L = 1.
  • Forgetting to check positivity in comparison tests is wrong because direct and limit comparison require eventually positive terms.
  • Using the alternating series test without checking that bnb_n decreases is wrong because both bn+1≤bnb_{n+1} \le b_n and lim⁡n→∞bn=0\lim_{n \to \infty} b_n = 0 are required.
  • Choosing direct comparison with the inequality in the wrong direction is wrong because an≤bna_n \le b_n only proves convergence from a larger convergent series, while an≥bna_n \ge b_n only proves divergence from a smaller divergent series.

Practice Questions

  1. 1 Determine whether ∑n=1∞3n5n\sum_{n=1}^{\infty} \frac{3^n}{5^n} converges, 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.
  3. 3 Determine whether ∑n=1∞n!4n\sum_{n=1}^{\infty} \frac{n!}{4^n} converges or diverges.
  4. 4 For the series ∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}, explain why it converges conditionally rather than absolutely.

Understanding Convergence Tests Decision Flowchart with Examples

A good test choice begins with noticing which part of a term controls its long-term size. Constants, lower-degree polynomial terms, and a finite number of early terms do not affect convergence. For a rational expression, focus on the highest powers in the numerator and denominator.

A term that behaves like one over n cubed has the same important behavior as a p-series with p equal to three. A term that behaves like n squared over n to the fifth behaves like one over n cubed. This idea prevents unnecessary algebra.

Comparison is most useful when an expression is messy but its size is easy to bound. For positive terms, use a known convergent series as an upper bound when trying to prove convergence. Use a known divergent series as a lower bound when trying to prove divergence.

Limit comparison is often faster for fractions involving polynomials, roots, or sums of powers. Divide by the simplest familiar series with the same dominant power of n. If the resulting limit is a positive finite number, details such as added constants no longer matter.

Factorials and exponentials usually point toward the ratio test because consecutive terms simplify dramatically. For example, each new factorial introduces a factor of n plus one, while powers introduce a fixed base. The root test is especially natural when the entire term is raised to the n power.

Both tests measure whether terms shrink at an exponential rate. Their result equal to one is not a failure in the work.

It means the test cannot decide, so a different feature of the series must guide the next step. P-series often produce this undecided result.

Alternating signs need careful treatment. First ignore the signs and examine the positive sizes of the terms. If that positive series converges, the original series converges absolutely.

Absolute convergence is stronger because rearranging the terms cannot change the sum. If the positive series diverges but the sizes decrease toward zero, the alternating series can still converge conditionally. This happens with the alternating harmonic series.

Its terms get small enough for cancellation to work, yet their absolute values do not form a convergent series. The error after stopping is no larger than the first omitted term.

The integral test gives more than a yes or no conclusion. It connects a discrete sum with area under a curve. When a positive decreasing function is used, the leftover tail of the series can be estimated with an improper integral.

This matters in applications where an infinite process is approximated by a finite calculation. In homework, write the test conditions before integrating.

Check positivity, continuity, and decrease on the relevant interval. A correct integral does not justify the test if those conditions are missing.