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 = 0.01

A ball is dropped and its first rebound reaches 20 m. Every next rebound reaches 80% of the previous one. Which rebound n is the first one lower than 0.01 m?

a₁ = 20r = 0.8ε = 0.01
n =
2.Easy
kak​123456789101112

Find the sum S19 of the first 19 terms of the arithmetic sequence 19, 20.5, 22, 23.5, …

a₁ = 19d = 1.5n = 19
S19 =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ k!/kᵏ. Find L = lim |ak+1/ak|.

ak = k!/kᵏ
L =
4.Medium
kSn​12345678910

Find the sum S = Σk=135 4/(k(k + 1)).

ak = 4/(k(k + 1))n = 35
S =
5.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 0.0001.

f(x) = cos xx = 1.5M = 1ε = 0.0001
n =
6.Hard

Find the coefficient c₂ of (x − 8)² in the Taylor series of f(x) = ∛x about a = 8.

f(x) = ∛xa = 8k = 2
c₂ =

Answer key

  1. 1. n = 36
  2. 2. S19 = 617.5
  3. 3. L = 0.3679
  4. 4. S = 3.889
  5. 5. n = 9
  6. 6. c₂ = −0.003472

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