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 = 0.2

The geometric sequence aₙ = 5·0.9ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.2.

a₁ = 5r = 0.9ε = 0.2
n =
2.Easy
kak​123456789101112

The arithmetic sequence 14, 14.5, 15, 15.5, … ends with the term 25.5. How many terms n does it have?

a₁ = 14d = 0.5aₙ = 25.5
n =
3.Medium
AΣ 1/(k² + 7)k = 1 … ∞BΣ k²/3ᵏk = 1 … ∞CΣ 2ᵏ/k³k = 1 … ∞DΣ (1/2)ᵏk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 1/(k² + 7)B: Σ k²/3ᵏC: Σ 2ᵏ/k³D: Σ (1/2)ᵏ
  1. 1A: Σ 1/(k² + 7)
  2. 2B: Σ k²/3ᵏ
  3. 3C: Σ 2ᵏ/k³
  4. 4D: Σ (1/2)ᵏ
4.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 4/(k(k + 2)).

ak = 4/(k(k + 2))
S =
5.Hard

f(x) = cos 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 0.001.

f(x) = cos xx = 2M = 1ε = 0.001
n =
6.Hard

f(x) = eˣ is approximated at x = 2 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 8, find the smallest degree n that guarantees an error below 0.0005.

f(x) = eˣx = 2M = 8ε = 0.0005
n =

Answer key

  1. 1. n = 32
  2. 2. n = 24
  3. 3. C: Σ 2ᵏ/k³
  4. 4. S = 3
  5. 5. n = 9
  6. 6. n = 10

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