doing math

courses › Calculus

Series

level 50 course

Infinite sums: when they add up, and to what.

Pen and paper is fine · no calculator needed why?

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Opens at level 50.

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Builds on: Integrals (not open yet) · Sequences & series (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

A series is an infinite sum. It converges when its running totals settle on one number, and diverges when they don't.

A power series has x in it and converges for x within some distance of its centre: the radius of convergence. A Maclaurin series writes a function as a power series centred at 0.

Techniques

Geometric: first term over 1 − r

A geometric series: each term is the one before times a fixed ratio r. Also a radius of convergence.

  1. r is the number raised to the power n.
  2. First term: put in the starting n.
  3. Sum = first term ÷ (1 − r), when r is between −1 and 1.
  4. Radius: the ratio must be below 1 in size, so (x/5)ⁿ gives 5 and (5x)ⁿ gives 0.2.
worked example

Example: What is Σ 2·(2/3)ⁿ for n = 1 to ∞? Enter a fraction or a whole number.

  1. Ratio 2/3. First term, at n = 1: 2 × 2/3 = 4/3.
  2. 1 − 2/3 = 1/3.
  3. (4/3) ÷ (1/3) = 4.

Answer: 4

Converge or diverge

Deciding whether a series settles on a number.

  1. Terms not shrinking to 0: diverges. Σ n/(n + 1) has terms heading to 1.
  2. Σ 1/nᵖ converges only when p > 1, so Σ 1/n and Σ 1/√n diverge.
  3. Positive terms below a convergent series’ terms converge, and so do terms shrinking steadily to 0 with alternating signs.
worked example

Example: Each sum runs from n = 1 to ∞. How many of these converge: Σ 1/n, Σ 1/n², Σ (3/2)ⁿ, Σ (1/2)ⁿ?

  1. Σ 1/n diverges (p is 1). Σ 1/n² converges (p is 2).
  2. Geometric: ratio 3/2 diverges, ratio 1/2 converges.
  3. Two converge.

Answer: 2

Read coefficients from known series

The coefficient of xᵏ in e^(ax), sin(ax), cos(ax), 1/(1 − ax) or ln(1 + ax).

  1. eᵘ = 1 + u + u²/2 + u³/6 + u⁴/24 + u⁵/120 + …; uᵏ is divided by 1 × 2 × … × k.
  2. sin u = u − u³/6 + u⁵/120 − …, and cos u = 1 − u²/2 + u⁴/24 − ….
  3. 1/(1 − u) = 1 + u + u² + …, and ln(1 + u) = u − u²/2 + u³/3 − ….
  4. Put u = ax. The xᵏ coefficient picks up aᵏ.
worked example

Example: What is the coefficient of x³ in the Maclaurin series of sin(2x)? Enter a fraction or a whole number.

  1. Put u = 2x into sin u = u − u³/6 + …
  2. The x³ term is −(2x)³/6, so the coefficient is −8/6 = −4/3.

Answer: −4/3

Tips by skill

  • TipGeometric series: Sum = first term ÷ (1 − ratio). Get the first term by putting in the starting value of n.
  • TipConverge or diverge?: Terms not shrinking to 0: diverges. Geometric: converges when the ratio is between −1 and 1. Σ 1/nᵖ: converges only when p > 1.
  • TipTaylor coefficients: Write the known series in u, then put u = ax. eᵘ, sin and cos divide uᵏ by 1 × 2 × … × k.
  • TipRadius of convergence: Make the term (ratio)ⁿ. Ratio x/a or (x − c)/a gives radius a; ratio ax gives 1/a, as a decimal. A factor n changes nothing.

Watch out for

  • Starting at the wrong term: from n = 1, the first term already includes one r.
  • Calling Σ 1/n convergent because its terms shrink to 0. It diverges, as do Σ 1/(2n) and Σ 1/(n + 1).
  • Dividing by k, not 1 × 2 × … × k, for eˣ, sin and cos. Only ln(1 + u) divides by plain k.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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