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 2 m. Every next rebound reaches 85% of the previous one. Which rebound n is the first one lower than 0.05 m?

a₁ = 2r = 0.85ε = 0.05
n =
2.Easy
kak​13579111315ε = 0.05

The geometric sequence aₙ = 2·0.9ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.05.

a₁ = 2r = 0.9ε = 0.05
n =
3.Medium
xx₀ = 1??R

The power series Σk=1∞ k!·(x − 1)ᵏ/kᵏ converges on an interval around x₀ = 1. Find the right end x of that interval.

Σ k!·(x − 1)ᵏ/kᵏx₀ = 1
x =
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ k³·2ᵏ/3ᵏ. Find L = lim |ak+1/ak|.

ak = k³·2ᵏ/3ᵏ
L =
5.Hard

f(x) = ln(1 + x) is approximated by its Maclaurin polynomial of degree 3 at x = 0.1. 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 = 0.1M = 6
|R₃| =
6.Hard

f(x) = cos 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 0.0005.

f(x) = cos xx = 1.5M = 1ε = 0.0005
n =

Answer key

  1. 1. n = 24
  2. 2. n = 37
  3. 3. x = 3.718
  4. 4. L = 0.6667
  5. 5. |R₃| = 2.5 × 10⁻⁵
  6. 6. n = 8

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