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 = 0.1

A ball is dropped and its first rebound reaches 10 m. Every next rebound reaches 70% of the previous one. Which rebound n is the first one lower than 0.1 m?

a₁ = 10r = 0.7ε = 0.1
n =
2.Easy
kak​123456789101112

Find the sum S23 of the first 23 terms of the arithmetic sequence 10, 6, 2, −2, …

a₁ = 10d = −4n = 23
S23 =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ k²·5ᵏ/3ᵏ. Find L = lim |ak+1/ak|.

ak = k²·5ᵏ/3ᵏ
L =
4.Medium
AΣ (−1)ᵏ/kk = 1 … ∞BΣ 1/(4k + 5)k = 1 … ∞CΣ (k + 4)/k²k = 1 … ∞DΣ (5/3)ᵏk = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ (−1)ᵏ/kB: Σ 1/(4k + 5)C: Σ (k + 4)/k²D: Σ (5/3)ᵏ
  1. 1A: Σ (−1)ᵏ/k
  2. 2B: Σ 1/(4k + 5)
  3. 3C: Σ (k + 4)/k²
  4. 4D: Σ (5/3)ᵏ
5.Hard

Recognise the series and find its sum: S = Σk=1∞ (−1)k+1·0.2ᵏ/k.

ak = (−1)k+1·0.2ᵏ/k
S =
6.Hard

f(x) = eˣ 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 = 2, find the smallest degree n that guarantees an error below 0.0001.

f(x) = eˣx = 0.5M = 2ε = 0.0001
n =

Answer key

  1. 1. n = 14
  2. 2. S23 = −782
  3. 3. L = 1.667
  4. 4. A: Σ (−1)ᵏ/k
  5. 5. S = 0.1823
  6. 6. n = 5

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