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

A ball is dropped and its first rebound reaches 2 m. Every next rebound reaches 90% of the previous one. Which rebound n is the first one lower than 0.1 m?

a₁ = 2r = 0.9ε = 0.1
n =
2.Easy
kak​135791113ε = 0.01

The geometric sequence aₙ = 20·0.5ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.01.

a₁ = 20r = 0.5ε = 0.01
n =
3.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 5/(k(k + 2)).

ak = 5/(k(k + 2))
S =
4.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 1/(k(k + 3)).

ak = 1/(k(k + 3))
S =
5.Hard

f(x) = eˣ is approximated by its Maclaurin polynomial of degree 5 at x = 1. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = e1 = 2.718 to find the largest possible error |R₅|.

f(x) = eˣn = 5x = 1M = 2.718
|R₅| =
6.Hard

f(x) = eˣ is approximated at x = 1 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 3, find the smallest degree n that guarantees an error below 1 × 10⁻⁵.

f(x) = eˣx = 1M = 3ε = 1 × 10⁻⁵
n =

Answer key

  1. 1. n = 30
  2. 2. n = 12
  3. 3. S = 3.75
  4. 4. S = 0.6111
  5. 5. |R₅| = 0.003775
  6. 6. n = 8

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