Sequences & series worksheet with answers

Free printable Sequences & series worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
kak​13579111315ε = 0.5

The geometric sequence aₙ = 20·0.85ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.5.

a₁ = 20r = 0.85ε = 0.5
n =
2.Easy
kak​123456789101112

The arithmetic sequence 7, 5, 3, 1, … ends with the term −17. How many terms n does it have?

a₁ = 7d = −2aₙ = −17
n =
3.Medium
AΣ 1/(2ᵏ + k)k = 1 … ∞BΣ 2/kk = 1 … ∞CΣ 1/k⁴k = 1 … ∞DΣ (2/5)ᵏk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 1/(2ᵏ + k)B: Σ 2/kC: Σ 1/k⁴D: Σ (2/5)ᵏ
  1. 1A: Σ 1/(2ᵏ + k)
  2. 2B: Σ 2/k
  3. 3C: Σ 1/k⁴
  4. 4D: Σ (2/5)ᵏ
4.Medium
AΣ (5/4)ᵏk = 1 … ∞BΣ 1/kk = 1 … ∞CΣ (3/5)ᵏk = 1 … ∞DΣ 2ᵏ/k³k = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ (5/4)ᵏB: Σ 1/kC: Σ (3/5)ᵏD: Σ 2ᵏ/k³
  1. 1A: Σ (5/4)ᵏ
  2. 2B: Σ 1/k
  3. 3C: Σ (3/5)ᵏ
  4. 4D: Σ 2ᵏ/k³
5.Hard

f(x) = sin x is approximated at x = 2 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 1, find the smallest degree n that guarantees an error below 1 × 10⁻⁵.

f(x) = sin xx = 2M = 1ε = 1 × 10⁻⁵
n =
6.Hard
xyfP₃0.5

Use the Maclaurin polynomial of degree 3 of f(x) = ln(1 − x) to approximate f(0.5). Give P₃(0.5).

f(x) = ln(1 − x)n = 3x = 0.5
P₃(0.5) =

Answer key

  1. 1. n = 24
  2. 2. n = 13
  3. 3. B: Σ 2/k
  4. 4. C: Σ (3/5)ᵏ
  5. 5. n = 11
  6. 6. P₃(0.5) = −0.6667

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