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 = 0.1

The geometric sequence aₙ = 10·0.9ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.1.

a₁ = 10r = 0.9ε = 0.1
n =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=1∞ 2·(4/5)ᵏ.

a = 2r = 4/5k₀ = 1
S =
3.Medium
AΣ k!/2ᵏk = 1 … ∞BΣ 1/(5k + 4)k = 1 … ∞CΣ (−1)ᵏ/√kk = 1 … ∞DΣ ln k / kk = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ k!/2ᵏB: Σ 1/(5k + 4)C: Σ (−1)ᵏ/√kD: Σ ln k / k
  1. 1A: Σ k!/2ᵏ
  2. 2B: Σ 1/(5k + 4)
  3. 3C: Σ (−1)ᵏ/√k
  4. 4D: Σ ln k / k
4.Medium
xx₀ = 0??R

Find the radius of convergence R of the power series Σk=1∞ 9ᵏx²ᵏ/k.

Σ 9ᵏx²ᵏ/kx₀ = 0
R =
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 0.0005.

f(x) = sin xx = 0.5M = 1ε = 0.0005
n =
6.Hard

Recognise the series and find its sum: S = Σk=1∞ (−1)k+1·0.5ᵏ/k.

ak = (−1)k+1·0.5ᵏ/k
S =

Answer key

  1. 1. n = 45
  2. 2. S = 8
  3. 3. C: Σ (−1)ᵏ/√k
  4. 4. R = 0.3333
  5. 5. n = 4
  6. 6. S = 0.4055

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