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​12345678ε = 0.5

A ball is dropped and its first rebound reaches 3 m. Every next rebound reaches 70% of the previous one. Which rebound n is the first one lower than 0.5 m?

a₁ = 3r = 0.7ε = 0.5
n =
2.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 =
3.Medium
AΣ 1/k²k = 1 … ∞BΣ (3/5)ᵏk = 1 … ∞CΣ 1/(k² + 7)k = 1 … ∞DΣ k!/10ᵏk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 1/k²B: Σ (3/5)ᵏC: Σ 1/(k² + 7)D: Σ k!/10ᵏ
  1. 1A: Σ 1/k²
  2. 2B: Σ (3/5)ᵏ
  3. 3C: Σ 1/(k² + 7)
  4. 4D: Σ k!/10ᵏ
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ (k!)²·5ᵏ/(2k)!. Find L = lim |ak+1/ak|.

ak = (k!)²·5ᵏ/(2k)!
L =
5.Hard

Recognise the series and find its sum: S = Σk=0∞ (−1)ᵏ·22k+1/(2k + 1)!.

ak = (−1)ᵏ·22k+1/(2k + 1)!
S =
6.Hard

Find the coefficient c₃ of (x − 1)³ in the Taylor series of f(x) = 1/x about a = 1.

f(x) = 1/xa = 1k = 3
c₃ =

Answer key

  1. 1. n = 7
  2. 2. n = 5
  3. 3. D: Σ k!/10ᵏ
  4. 4. L = 1.25
  5. 5. S = 0.9093
  6. 6. c₃ = −1

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