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​123456ε = 0.2

The geometric sequence aₙ = 3·0.5ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.2.

a₁ = 3r = 0.5ε = 0.2
n =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=0∞ 10·(−1/3)ᵏ.

a = 10r = −1/3k₀ = 0
S =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

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

ak = (k!)²·2ᵏ/(2k)!
L =
4.Medium
AΣ (2/3)ᵏk = 1 … ∞BΣ (1 + 1/k)ᵏk = 1 … ∞CΣ (7/4)ᵏk = 1 … ∞DΣ (k + 2)/k²k = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ (2/3)ᵏB: Σ (1 + 1/k)ᵏC: Σ (7/4)ᵏD: Σ (k + 2)/k²
  1. 1A: Σ (2/3)ᵏ
  2. 2B: Σ (1 + 1/k)ᵏ
  3. 3C: Σ (7/4)ᵏ
  4. 4D: Σ (k + 2)/k²
5.Hard

f(x) = sin x is approximated at x = 1 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 1 × 10⁻⁶.

f(x) = sin xx = 1M = 1ε = 1 × 10⁻⁶
n =
6.Hard

Recognise the series and find its sum: S = Σk=1∞ k·0.2ᵏ.

ak = k·0.2ᵏ
S =

Answer key

  1. 1. n = 5
  2. 2. S = 7.5
  3. 3. L = 0.5
  4. 4. A: Σ (2/3)ᵏ
  5. 5. n = 9
  6. 6. S = 0.3125

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