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Series

lesson · about 3 minutes

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

Pen and paper is fine · no calculator needed why?

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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

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Geometric series

worked example

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

Answer: -2/3

  1. It is geometric: first term −1 (at n = 1), ratio −1/2. The sum is first term ÷ (1 − ratio).
  2. (−1) ÷ (1 − (−1/2)) = (−1) ÷ (3/2) = −2/3.

Converge or diverge?

worked example

Which series diverges? Each sum runs from n = 1 to ∞.

  1. Σ n/2ⁿ
  2. Σ √n
  3. Σ (2/3)ⁿ
  4. Σ 1/n²

Answer: Σ √n

  1. Σ √n diverges: its terms grow instead of shrinking to 0.
  2. The others converge: Σ n/2ⁿ (ratio test gives 1/2), Σ (2/3)ⁿ (geometric, ratio 2/3), Σ 1/n² (p = 2 > 1).

Taylor coefficients

worked example

What is the coefficient of x² in the Maclaurin series of e⁻³ˣ? Enter a fraction or a whole number.

Answer: 9/2

  1. e⁻³ˣ = Σ (−3x)ⁿ/n!, so the x² term has coefficient (−3)²/2! = 9/2.

Radius of convergence

worked example

What is the radius of convergence of Σ xⁿ/3ⁿ (n from 0 to ∞)? Enter a whole number or a decimal.

Answer: 3

  1. It is geometric with ratio x/3, which converges when |x/3| < 1, that is |x| < 3.
  2. Radius 3.

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