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 = 20·0.75ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.1.

a₁ = 20r = 0.75ε = 0.1
n =
2.Easy
kak​123456789

Find the term a9 of the arithmetic sequence −8, −6, −4, −2, …

a₁ = −8d = 2n = 9
a9 =
3.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 12/(k(k + 2)).

ak = 12/(k(k + 2))
S =
4.Medium
xx₀ = 0??R

Find the radius of convergence R of the power series Σk=1∞ k!·xᵏ/kᵏ.

Σ k!·xᵏ/kᵏx₀ = 0
R =
5.Hard

f(x) = cos x is approximated by its Maclaurin polynomial of degree 3 at x = 0.8. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₃|.

f(x) = cos xn = 3x = 0.8M = 1
|R₃| =
6.Hard

f(x) = eˣ 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 = 2, find the smallest degree n that guarantees an error below 0.001.

f(x) = eˣx = 0.5M = 2ε = 0.001
n =

Answer key

  1. 1. n = 20
  2. 2. a9 = 8
  3. 3. S = 9
  4. 4. R = 2.718
  5. 5. |R₃| = 0.01707
  6. 6. n = 4

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