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
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=1∞ 2·(−3/4)ᵏ.

a = 2r = −3/4k₀ = 1
S =
2.Easy
kak​123456789101112

Find the term a37 of the arithmetic sequence 12, 14, 16, 18, …

a₁ = 12d = 2n = 37
a37 =
3.Medium
k|aₖ|¹ᐟᵏ123456789101112

Apply the root test to Σk=1∞ ((1k + 3)/(3k + 3))ᵏ. Find L = lim |ak|1/k.

ak = ((1k + 3)/(3k + 3))ᵏ
L =
4.Medium
AΣ (3/5)ᵏk = 1 … ∞BΣ (4k + 1)/(k + 2)k = 1 … ∞CΣ (3/2)ᵏk = 1 … ∞DΣ ln k / kk = 1 … ∞

Exactly one of the four series converges. Click it.

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

f(x) = cos 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 1 × 10⁻⁶.

f(x) = cos xx = 0.5M = 1ε = 1 × 10⁻⁶
n =
6.Hard

Recognise the series and find its sum: S = Σk=1∞ (−1)k+1·0.25ᵏ/k.

ak = (−1)k+1·0.25ᵏ/k
S =

Answer key

  1. 1. S = −0.8571
  2. 2. a37 = 84
  3. 3. L = 0.3333
  4. 4. A: Σ (3/5)ᵏ
  5. 5. n = 7
  6. 6. S = 0.2231

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