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 = 0.1

The geometric sequence aₙ = 2·0.85ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.1.

a₁ = 2r = 0.85ε = 0.1
n =
2.Easy
kak​123456789101112ε = 0.2

The geometric sequence aₙ = 20·0.6ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.2.

a₁ = 20r = 0.6ε = 0.2
n =
3.Medium
xx₀ = 0??R

Find the radius of convergence R of the power series Σk=1∞ 4ᵏx²ᵏ/k.

Σ 4ᵏx²ᵏ/kx₀ = 0
R =
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ (k!)²·5ᵏ/(2k)!. Find L = lim |ak+1/ak|.

ak = (k!)²·5ᵏ/(2k)!
L =
5.Hard

Recognise the series and find its sum: S = Σk=0∞ 1.5ᵏ/k!.

ak = 1.5ᵏ/k!
S =
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 1 × 10⁻⁵.

f(x) = sin xx = 1M = 1ε = 1 × 10⁻⁵
n =

Answer key

  1. 1. n = 20
  2. 2. n = 11
  3. 3. R = 0.5
  4. 4. L = 1.25
  5. 5. S = 4.482
  6. 6. n = 8

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