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∞ 3·(3/5)ᵏ.

a = 3r = 3/5k₀ = 1
S =
2.Easy
kak​123456789101112ε = 0.2

The geometric sequence aₙ = 3·0.75ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.2.

a₁ = 3r = 0.75ε = 0.2
n =
3.Medium
kSn​12345678910

Find the sum S = Σk=131 2/(k(k + 1)).

ak = 2/(k(k + 1))n = 31
S =
4.Medium
AΣ 2ᵏ/k³k = 1 … ∞BΣ (1/2)ᵏk = 1 … ∞CΣ 3ᵏ/k!k = 1 … ∞DΣ (−1)ᵏ/kk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 2ᵏ/k³B: Σ (1/2)ᵏC: Σ 3ᵏ/k!D: Σ (−1)ᵏ/k
  1. 1A: Σ 2ᵏ/k³
  2. 2B: Σ (1/2)ᵏ
  3. 3C: Σ 3ᵏ/k!
  4. 4D: Σ (−1)ᵏ/k
5.Hard

Recognise the series and find its sum: S = Σk=0∞ (−2)ᵏ/k!.

ak = (−2)ᵏ/k!
S =
6.Hard

f(x) = cos x is approximated at x = 1.5 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 = 1.5M = 1ε = 1 × 10⁻⁶
n =

Answer key

  1. 1. S = 4.5
  2. 2. n = 11
  3. 3. S = 1.938
  4. 4. A: Σ 2ᵏ/k³
  5. 5. S = 0.1353
  6. 6. n = 11

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