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
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=1∞ 5·(2/5)ᵏ.

a = 5r = 2/5k₀ = 1
S =
2.Easy
kak​12345

Find the sum S5 of the first 5 terms of the geometric sequence 0.5, −0.25, 0.125, …

a₁ = 0.5r = −0.5n = 5
S5 =
3.Medium
k|aₖ|¹ᐟᵏ123456789101112

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

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

Find the sum S = Σk=126 4/(k(k + 1)).

ak = 4/(k(k + 1))n = 26
S =
5.Hard

f(x) = sin 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 × 10⁻⁶.

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

Find the coefficient c₂ of (x − 4)² in the Taylor series of f(x) = ln x about a = 4.

f(x) = ln xa = 4k = 2
c₂ =

Answer key

  1. 1. S = 3.333
  2. 2. S5 = 0.3438
  3. 3. L = 0.8
  4. 4. S = 3.852
  5. 5. n = 7
  6. 6. c₂ = −0.03125

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