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​1234567891011ε = 0.5

The geometric sequence aₙ = 5·0.75ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.5.

a₁ = 5r = 0.75ε = 0.5
n =
2.Easy
kak​123456789101112

Find the sum S20 of the first 20 terms of the arithmetic sequence 4, 9, 14, 19, …

a₁ = 4d = 5n = 20
S20 =
3.Medium
AΣ (2/3)ᵏk = 1 … ∞BΣ k²/2ᵏk = 1 … ∞CΣ (5/3)ᵏk = 1 … ∞DΣ 1/(k(k + 1))k = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ (2/3)ᵏB: Σ k²/2ᵏC: Σ (5/3)ᵏD: Σ 1/(k(k + 1))
  1. 1A: Σ (2/3)ᵏ
  2. 2B: Σ k²/2ᵏ
  3. 3C: Σ (5/3)ᵏ
  4. 4D: Σ 1/(k(k + 1))
4.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 12/(k(k + 1)).

ak = 12/(k(k + 1))
S =
5.Hard

f(x) = eˣ is approximated at x = 1 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 3, find the smallest degree n that guarantees an error below 0.001.

f(x) = eˣx = 1M = 3ε = 0.001
n =
6.Hard

f(x) = sin x is approximated at x = 1 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 0.0001.

f(x) = sin xx = 1M = 1ε = 0.0001
n =

Answer key

  1. 1. n = 10
  2. 2. S20 = 1030
  3. 3. C: Σ (5/3)ᵏ
  4. 4. S = 12
  5. 5. n = 6
  6. 6. n = 7

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