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 = 0.01

The geometric sequence aₙ = 3·0.6ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.01.

a₁ = 3r = 0.6ε = 0.01
n =
2.Easy
kak​12345678910ε = 0.5

The geometric sequence aₙ = 8·0.7ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.5.

a₁ = 8r = 0.7ε = 0.5
n =
3.Medium
AΣ 3ᵏ/k!k = 1 … ∞BΣ 1/(2ᵏ + k)k = 1 … ∞CΣ (k + 2)/k²k = 1 … ∞DΣ (2/5)ᵏk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 3ᵏ/k!B: Σ 1/(2ᵏ + k)C: Σ (k + 2)/k²D: Σ (2/5)ᵏ
  1. 1A: Σ 3ᵏ/k!
  2. 2B: Σ 1/(2ᵏ + k)
  3. 3C: Σ (k + 2)/k²
  4. 4D: Σ (2/5)ᵏ
4.Medium
xx₀ = 0??R

Find the radius of convergence R of the power series Σk=1∞ 4ᵏx²ᵏ/k.

Σ 4ᵏx²ᵏ/kx₀ = 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) = sin x is approximated by its Maclaurin polynomial of degree 5 at x = 0.3. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₅|.

f(x) = sin xn = 5x = 0.3M = 1
|R₅| =

Answer key

  1. 1. n = 13
  2. 2. n = 9
  3. 3. C: Σ (k + 2)/k²
  4. 4. R = 0.5
  5. 5. P₃(0.25) = 0.4167
  6. 6. |R₅| = 1.012 × 10⁻⁶

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