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 = 0.2

The geometric sequence aₙ = 10·0.5ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.2.

a₁ = 10r = 0.5ε = 0.2
n =
2.Easy
kak​1234567

Find the sum S7 of the first 7 terms of the geometric sequence 4, 8, 16, …

a₁ = 4r = 2n = 7
S7 =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

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

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

Exactly one of the four series diverges. Click it.

A: Σ 5/k²B: Σ 1/kC: Σ (−1)ᵏ/√kD: Σ k²/5ᵏ
  1. 1A: Σ 5/k²
  2. 2B: Σ 1/k
  3. 3C: Σ (−1)ᵏ/√k
  4. 4D: Σ k²/5ᵏ
5.Hard

f(x) = cos x is approximated at x = 1 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 0.0005.

f(x) = cos xx = 1M = 1ε = 0.0005
n =
6.Hard

Find the coefficient c₄ of (x − 2)⁴ in the Taylor series of f(x) = ln x about a = 2.

f(x) = ln xa = 2k = 4
c₄ =

Answer key

  1. 1. n = 7
  2. 2. S7 = 508
  3. 3. L = 1.104
  4. 4. B: Σ 1/k
  5. 5. n = 6
  6. 6. c₄ = −0.01563

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