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

Find the sum S33 of the first 33 terms of the arithmetic sequence 12, 10, 8, 6, …

a₁ = 12d = −2n = 33
S33 =
2.Easy
kak​13579111315ε = 0.1

The geometric sequence aₙ = 5·0.8ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.1.

a₁ = 5r = 0.8ε = 0.1
n =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ (k!)²·8ᵏ/(2k)!. Find L = lim |ak+1/ak|.

ak = (k!)²·8ᵏ/(2k)!
L =
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ kᵏ/(3ᵏ·k!). Find L = lim |ak+1/ak|.

ak = kᵏ/(3ᵏ·k!)
L =
5.Hard

f(x) = sin x is approximated at x = 1 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 = 1M = 1ε = 0.0005
n =
6.Hard

Find the coefficient c₂ of (x − 1)² in the Taylor series of f(x) = ∛x about a = 1.

f(x) = ∛xa = 1k = 2
c₂ =

Answer key

  1. 1. S33 = −660
  2. 2. n = 19
  3. 3. L = 2
  4. 4. L = 0.9061
  5. 5. n = 6
  6. 6. c₂ = −0.1111

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