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
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Find the sum S38 of the first 38 terms of the arithmetic sequence −5, −2, 1, 4, …

a₁ = −5d = 3n = 38
S38 =
2.Easy
kak​13579111315ε = 0.1

A ball is dropped and its first rebound reaches 8 m. Every next rebound reaches 85% of the previous one. Which rebound n is the first one lower than 0.1 m?

a₁ = 8r = 0.85ε = 0.1
n =
3.Medium
AΣ 1/(k(k + 2))k = 1 … ∞BΣ k/(k + 9)k = 1 … ∞CΣ 5/k²k = 1 … ∞DΣ k³/3ᵏk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 1/(k(k + 2))B: Σ k/(k + 9)C: Σ 5/k²D: Σ k³/3ᵏ
  1. 1A: Σ 1/(k(k + 2))
  2. 2B: Σ k/(k + 9)
  3. 3C: Σ 5/k²
  4. 4D: Σ k³/3ᵏ
4.Medium
AΣ 3ᵏ/k²k = 1 … ∞BΣ 5/k²k = 1 … ∞CΣ 1/(4k + 1)k = 1 … ∞DΣ (5/4)ᵏk = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ 3ᵏ/k²B: Σ 5/k²C: Σ 1/(4k + 1)D: Σ (5/4)ᵏ
  1. 1A: Σ 3ᵏ/k²
  2. 2B: Σ 5/k²
  3. 3C: Σ 1/(4k + 1)
  4. 4D: Σ (5/4)ᵏ
5.Hard

f(x) = ln(1 + x) is approximated by its Maclaurin polynomial of degree 4 at x = 0.2. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 4! = 24 to find the largest possible error |R₄|.

f(x) = ln(1 + x)n = 4x = 0.2M = 24
|R₄| =
6.Hard

f(x) = sin x is approximated by its Maclaurin polynomial of degree 4 at x = 0.8. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₄|.

f(x) = sin xn = 4x = 0.8M = 1
|R₄| =

Answer key

  1. 1. S38 = 1919
  2. 2. n = 28
  3. 3. B: Σ k/(k + 9)
  4. 4. B: Σ 5/k²
  5. 5. |R₄| = 6.4 × 10⁻⁵
  6. 6. |R₄| = 0.002731

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