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

The geometric sequence aₙ = 10·0.6ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.01.

a₁ = 10r = 0.6ε = 0.01
n =
2.Easy
kak​123456789101112

Find the sum S36 of the first 36 terms of the arithmetic sequence 3, 0, −3, −6, …

a₁ = 3d = −3n = 36
S36 =
3.Medium
k|aₖ|¹ᐟᵏ123456789101112

Apply the root test to Σk=1∞ ((4k + 6)/(5k + 1))ᵏ. Find L = lim |ak|1/k.

ak = ((4k + 6)/(5k + 1))ᵏ
L =
4.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 12/(k(k + 4)).

ak = 12/(k(k + 4))
S =
5.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 1 × 10⁻⁶.

f(x) = eˣx = 0.5M = 2ε = 1 × 10⁻⁶
n =
6.Hard

Find the coefficient c₃ of (x − 4)³ in the Taylor series of f(x) = √x about a = 4.

f(x) = √xa = 4k = 3
c₃ =

Answer key

  1. 1. n = 15
  2. 2. S36 = −1782
  3. 3. L = 0.8
  4. 4. S = 6.25
  5. 5. n = 7
  6. 6. c₃ = 0.001953

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