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 = 0.05

A ball is dropped and its first rebound reaches 5 m. Every next rebound reaches 80% of the previous one. Which rebound n is the first one lower than 0.05 m?

a₁ = 5r = 0.8ε = 0.05
n =
2.Easy
kak​123456789101112

The arithmetic sequence −7, −11, −15, −19, … ends with the term −147. How many terms n does it have?

a₁ = −7d = −4aₙ = −147
n =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

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

ak = (k!)²·1ᵏ/(2k)!
L =
4.Medium
AΣ (3/5)ᵏk = 1 … ∞BΣ 1/(k² + 5)k = 1 … ∞CΣ (1 + 1/k)ᵏk = 1 … ∞DΣ (−1)ᵏ/kk = 1 … ∞

Exactly one of the four series diverges. Click it.

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

Recognise the series and find its sum: S = Σk=1∞ k·(1/3)ᵏ.

ak = k·(1/3)ᵏ
S =
6.Hard

f(x) = sin x is approximated by its Maclaurin polynomial of degree 4 at x = 0.2. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₄|.

f(x) = sin xn = 4x = 0.2M = 1
|R₄| =

Answer key

  1. 1. n = 22
  2. 2. n = 36
  3. 3. L = 0.25
  4. 4. C: Σ (1 + 1/k)ᵏ
  5. 5. S = 0.75
  6. 6. |R₄| = 2.667 × 10⁻⁶

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