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​123456789101112

The arithmetic sequence 12, 8, 4, 0, … ends with the term −140. How many terms n does it have?

a₁ = 12d = −4aₙ = −140
n =
2.Easy
kak​13579111315ε = 0.01

A ball is dropped and its first rebound reaches 8 m. Every next rebound reaches 85% of the previous one. Which rebound n is the first one lower than 0.01 m?

a₁ = 8r = 0.85ε = 0.01
n =
3.Medium
xx₀ = 0??R

Find the radius of convergence R of the power series Σk=1∞ 25ᵏx²ᵏ/k.

Σ 25ᵏx²ᵏ/kx₀ = 0
R =
4.Medium
k|aₖ|¹ᐟᵏ123456789101112

Apply the root test to Σk=1∞ ((2k + 6)/(4k + 3))ᵏ. Find L = lim |ak|1/k.

ak = ((2k + 6)/(4k + 3))ᵏ
L =
5.Hard

f(x) = cos 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 1 × 10⁻⁵.

f(x) = cos xx = 0.5M = 1ε = 1 × 10⁻⁵
n =
6.Hard

f(x) = cos x is approximated at x = 1.5 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 1 × 10⁻⁵.

f(x) = cos xx = 1.5M = 1ε = 1 × 10⁻⁵
n =

Answer key

  1. 1. n = 39
  2. 2. n = 43
  3. 3. R = 0.2
  4. 4. L = 0.5
  5. 5. n = 6
  6. 6. n = 10

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