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​1234567ε = 0.1

A ball is dropped and its first rebound reaches 2 m. Every next rebound reaches 50% of the previous one. Which rebound n is the first one lower than 0.1 m?

a₁ = 2r = 0.5ε = 0.1
n =
2.Easy
kak​135791113ε = 0.01

A ball is dropped and its first rebound reaches 2 m. Every next rebound reaches 60% of the previous one. Which rebound n is the first one lower than 0.01 m?

a₁ = 2r = 0.6ε = 0.01
n =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ k!/kᵏ. Find L = lim |ak+1/ak|.

ak = k!/kᵏ
L =
4.Medium
AΣ (3/4)ᵏk = 1 … ∞BΣ (−1)ᵏ/√kk = 1 … ∞CΣ (5/3)ᵏk = 1 … ∞DΣ 2/k²k = 1 … ∞

Exactly one of the four series diverges. Click it.

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

f(x) = ln(1 + x) is approximated by its Maclaurin polynomial of degree 3 at x = 0.5. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 3! = 6 to find the largest possible error |R₃|.

f(x) = ln(1 + x)n = 3x = 0.5M = 6
|R₃| =
6.Hard
xyfP₄1.5

Use the Maclaurin polynomial of degree 4 of f(x) = cos(−x) to approximate f(1.5). Give P₄(1.5).

f(x) = cos(−x)n = 4x = 1.5
P₄(1.5) =

Answer key

  1. 1. n = 6
  2. 2. n = 12
  3. 3. L = 0.3679
  4. 4. C: Σ (5/3)ᵏ
  5. 5. |R₃| = 0.01563
  6. 6. P₄(1.5) = 0.08594

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