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 = 20·0.9ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.1.

a₁ = 20r = 0.9ε = 0.1
n =
2.Easy
kak​123456789101112

The arithmetic sequence −8, −11, −14, −17, … ends with the term −86. How many terms n does it have?

a₁ = −8d = −3aₙ = −86
n =
3.Medium
xx₀ = 0??R

Find the radius of convergence R of the power series Σk=1∞ 9ᵏx²ᵏ/k.

Σ 9ᵏx²ᵏ/kx₀ = 0
R =
4.Medium
xx₀ = 2??R

The power series Σk=1∞ (2k)!·(x − 2)ᵏ/(k!)² converges on an interval around x₀ = 2. Find the left end x of that interval.

Σ (2k)!·(x − 2)ᵏ/(k!)²x₀ = 2
x =
5.Hard

Recognise the series and find its sum: S = Σk=0∞ 2ᵏ/k!.

ak = 2ᵏ/k!
S =
6.Hard

f(x) = sin x is approximated at x = 0.5 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 0.0005.

f(x) = sin xx = 0.5M = 1ε = 0.0005
n =

Answer key

  1. 1. n = 52
  2. 2. n = 27
  3. 3. R = 0.3333
  4. 4. x = 1.75
  5. 5. S = 7.389
  6. 6. n = 4

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