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Series and Sequences cheat sheet - grade 11-12

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Calculus Grade 11-12

Series and Sequences Cheat Sheet

A printable reference covering sequence notation, arithmetic and geometric series, convergence tests, power series, and Taylor series for grades 11-12.

TopicsCalculusDerivativesFunctionsAlgebraExponentsPatternsStudy Guide
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Calculus: Series and Sequences covers how ordered lists of numbers and infinite sums behave. Students need this cheat sheet because series questions often require choosing the right test before doing any algebra. It helps connect patterns, limits, convergence, and function approximations in one printable reference.

These ideas are essential for AP Calculus BC and first-year college calculus.

Key Facts

  • A sequence {an}\{a_n\} converges to LL if lim⁡n→∞an=L\lim_{n \to \infty} a_n = L.
  • An infinite 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 has a finite limit.
  • The 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| \ge 1.
  • The pp-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 divergence test says that 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.
  • 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.
  • A power series ∑n=0∞cn(x−a)n\sum_{n=0}^{\infty} c_n(x-a)^n converges for ∣x−a∣<R|x-a| < R, diverges for ∣x−a∣>R|x-a| > R, and must be checked separately at x=a±Rx=a \pm R.
  • The Taylor series for a function centered at aa is ∑n=0∞f(n)(a)n!(x−a)n\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n.

Vocabulary

Sequence
A sequence is an ordered list of numbers written as {an}\{a_n\}, where nn usually represents a positive integer position.
Series
A series is the sum of the terms of a sequence, 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 whether an infinite series converges.
Convergence
Convergence means a sequence or series approaches a finite value as n→∞n \to \infty.
Radius of Convergence
The radius of convergence RR is the distance from the center aa within which a power series ∑cn(x−a)n\sum c_n(x-a)^n converges.
Taylor Series
A Taylor series represents a function near x=ax=a using derivatives in the form ∑n=0∞f(n)(a)n!(x−a)n\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n.

Common Mistakes to Avoid

  • Using the divergence test to prove convergence is wrong because the test only proves divergence when lim⁡n→∞an≠0\lim_{n \to \infty} a_n \ne 0 or does not exist.
  • Forgetting endpoint checks for a power series is wrong because the ratio or root test usually gives only ∣x−a∣<R|x-a| < R, while x=a−Rx=a-R and x=a+Rx=a+R can behave differently.
  • Treating every alternating series as convergent is wrong because the alternating series test also requires bn→0b_n \to 0 and the positive terms bnb_n to eventually decrease.
  • Applying the geometric sum formula when ∣r∣≥1|r| \ge 1 is wrong because ∑n=0∞arn=a1−r\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} only holds for ∣r∣<1|r| < 1.
  • Ignoring absolute convergence is wrong because a series may converge conditionally, and tests like the ratio test often determine whether ∑∣an∣\sum |a_n| converges.

Practice Questions

  1. 1 Determine whether ∑n=1∞3n2\sum_{n=1}^{\infty} \frac{3}{n^2} converges or diverges, and name the test or rule used.
  2. 2 Find the sum of the geometric series ∑n=0∞5(23)n\sum_{n=0}^{\infty} 5\left(\frac{2}{3}\right)^n.
  3. 3 Find the radius and interval of convergence for ∑n=1∞(x−2)nn3n\sum_{n=1}^{\infty} \frac{(x-2)^n}{n3^n}.
  4. 4 Explain why checking lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0 is not enough to conclude that ∑n=1∞an\sum_{n=1}^{\infty} a_n converges.

Understanding Series and Sequences

A sequence is best viewed as a process that produces one value at each step. A series is what happens when those values are accumulated. This distinction matters because terms can become tiny without their total settling down.

For example, repeatedly adding positive amounts that shrink slowly can still produce an ever growing total. Partial sums make this visible. Write down the total after one term, then two terms, then three.

If those running totals crowd toward one fixed number, the infinite addition has a meaningful finite value. If they keep rising, falling, or jumping around, it does not.

A good test choice begins with the shape of the terms. First inspect the individual term far out in the sequence. If it fails to approach zero, no further convergence test can save the series.

This is a fast way to eliminate many problems. Next look for familiar structures. Constant ratios suggest a geometric pattern.

Powers of the index often suggest comparison with a p series. Factorials, products, or terms raised to the index usually make the ratio test efficient.

Comparison tests are especially useful when every term is positive. Instead of finding an exact sum, students compare the terms with a simpler benchmark whose behavior is already known.

Some series contain positive and negative terms that alternate. The signs can create cancellation, so their behavior may differ from the series made by taking every term as positive. This leads to an important distinction.

Absolute convergence means the total remains finite even after all signs are removed. Conditional convergence means cancellation is essential. In a conditionally convergent series, changing the order of addition can change the result.

This seems strange because ordinary finite sums can be reordered freely. Infinite processes need more care, since the idea of adding forever is defined through partial sums in a particular order.

Power series turn a list of coefficients into a function. Near the center, powers of the distance from that center become small, so later terms may have little effect. Farther away, those powers can grow and the same expression may stop working.

The boundary points deserve separate attention because they are not decided by the radius alone. Taylor series use derivatives to choose coefficients that match a function at one point. The constant term matches the height.

The next term matches the slope. Higher terms match bending and more detailed local behavior. This is why calculators, computer graphics, and physics models can replace difficult functions with polynomials over a limited range.

When working problems, track where the index starts and write several early terms before applying a test. A shift from zero to one can change the first term and the final value of a finite partial sum. In Taylor work, compute derivatives carefully and evaluate each one at the center before building terms.

Factorials grow very quickly, so missing one factorial changes the pattern substantially. Finally, separate two jobs in your written work. One job is proving convergence or divergence.

The other is finding a sum or approximation when one exists. A convergence test often answers only the first job.