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 = 0.05

The geometric sequence aₙ = 3·0.75ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.05.

a₁ = 3r = 0.75ε = 0.05
n =
2.Easy
kak​1234567

Find the sum S7 of the first 7 terms of the geometric sequence 2, −6, 18, …

a₁ = 2r = −3n = 7
S7 =
3.Medium
xx₀ = −1??R

Find the radius of convergence R of the power series Σk=1∞ (x + 1)ᵏ/(k·4ᵏ).

Σ (x + 1)ᵏ/(k·4ᵏ)x₀ = −1
R =
4.Medium
xx₀ = −1??R

The power series Σk=1∞ k!·(x + 1)ᵏ/kᵏ converges on an interval around x₀ = −1. Find the left end x of that interval.

Σ k!·(x + 1)ᵏ/kᵏx₀ = −1
x =
5.Hard
xyfP₄0.5

Use the Maclaurin polynomial of degree 4 of f(x) = 1/(1 − x) to approximate f(0.5). Give P₄(0.5).

f(x) = 1/(1 − x)n = 4x = 0.5
P₄(0.5) =
6.Hard

f(x) = sin x is approximated by its Maclaurin polynomial of degree 2 at x = −0.5. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₂|.

f(x) = sin xn = 2x = −0.5M = 1
|R₂| =

Answer key

  1. 1. n = 16
  2. 2. S7 = 1094
  3. 3. R = 4
  4. 4. x = −3.718
  5. 5. P₄(0.5) = 1.938
  6. 6. |R₂| = 0.02083

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