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 = 0.01

A ball is dropped and its first rebound reaches 10 m. Every next rebound reaches 85% of the previous one. Which rebound n is the first one lower than 0.01 m?

a₁ = 10r = 0.85ε = 0.01
n =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=2∞ 18·(−2/5)ᵏ.

a = 18r = −2/5k₀ = 2
S =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ kᵏ/(3ᵏ·k!). Find L = lim |ak+1/ak|.

ak = kᵏ/(3ᵏ·k!)
L =
4.Medium
AΣ 1/(3ᵏ + k)k = 1 … ∞BΣ 1/k³k = 1 … ∞CΣ (−1)ᵏ/√kk = 1 … ∞DΣ 2ᵏ/k²k = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 1/(3ᵏ + k)B: Σ 1/k³C: Σ (−1)ᵏ/√kD: Σ 2ᵏ/k²
  1. 1A: Σ 1/(3ᵏ + k)
  2. 2B: Σ 1/k³
  3. 3C: Σ (−1)ᵏ/√k
  4. 4D: Σ 2ᵏ/k²
5.Hard

f(x) = sin 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 0.001.

f(x) = sin xx = 1.5M = 1ε = 0.001
n =
6.Hard
xyfP₄2.5

Use the Maclaurin polynomial of degree 4 of f(x) = eˣ to approximate f(2.5). Give P₄(2.5).

f(x) = eˣn = 4x = 2.5
P₄(2.5) =

Answer key

  1. 1. n = 44
  2. 2. S = 2.057
  3. 3. L = 0.9061
  4. 4. D: Σ 2ᵏ/k²
  5. 5. n = 7
  6. 6. P₄(2.5) = 10.86

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