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​135791113ε = 0.1

The geometric sequence aₙ = 20·0.6ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.1.

a₁ = 20r = 0.6ε = 0.1
n =
2.Easy
kak​123456

Find the sum S6 of the first 6 terms of the geometric sequence 4, −12, 36, …

a₁ = 4r = −3n = 6
S6 =
3.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 =
4.Medium
xx₀ = 0??R

Find the radius of convergence R of the power series Σk=1∞ k!·xᵏ/kᵏ.

Σ k!·xᵏ/kᵏx₀ = 0
R =
5.Hard
xyfP₃0.25

Use the Maclaurin polynomial of degree 3 of f(x) = ln(1 + 2x) to approximate f(0.25). Give P₃(0.25).

f(x) = ln(1 + 2x)n = 3x = 0.25
P₃(0.25) =
6.Hard

f(x) = eˣ is approximated at x = 2 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 8, find the smallest degree n that guarantees an error below 1 × 10⁻⁵.

f(x) = eˣx = 2M = 8ε = 1 × 10⁻⁵
n =

Answer key

  1. 1. n = 12
  2. 2. S6 = −728
  3. 3. L = 0.9061
  4. 4. R = 2.718
  5. 5. P₃(0.25) = 0.4167
  6. 6. n = 13

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