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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Name: ____________________   Date: ____________

1.Easy
kak​123456789101112

Find the term a12 of the arithmetic sequence −1, 5, 11, 17, …

a₁ = −1d = 6n = 12
a12 =
2.Easy
kak​123456

Find the sum S6 of the first 6 terms of the geometric sequence 0.5, 0.25, 0.125, …

a₁ = 0.5r = 0.5n = 6
S6 =
3.Medium
AΣ (4/3)ᵏk = 1 … ∞BΣ (−1)ᵏ/kk = 1 … ∞CΣ ln k / kk = 1 … ∞DΣ 1/√kk = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ (4/3)ᵏB: Σ (−1)ᵏ/kC: Σ ln k / kD: Σ 1/√k
  1. 1A: Σ (4/3)ᵏ
  2. 2B: Σ (−1)ᵏ/k
  3. 3C: Σ ln k / k
  4. 4D: Σ 1/√k
4.Medium
k|aₖ₊₁/aₖ|123456789101112

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

ak = 2ᵏ·k!/kᵏ
L =
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 0.001.

f(x) = sin xx = 0.5M = 1ε = 0.001
n =
6.Hard

Find the coefficient c₄ of (x + 1)⁴ in the Taylor series of f(x) = eˣ about a = −1.

f(x) = eˣa = −1k = 4
c₄ =

Answer key

  1. 1. a12 = 65
  2. 2. S6 = 0.9844
  3. 3. B: Σ (−1)ᵏ/k
  4. 4. L = 0.7358
  5. 5. n = 4
  6. 6. c₄ = 0.01533

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