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 = 0.5

A ball is dropped and its first rebound reaches 20 m. Every next rebound reaches 75% of the previous one. Which rebound n is the first one lower than 0.5 m?

a₁ = 20r = 0.75ε = 0.5
n =
2.Easy
kak​1234567891011ε = 0.5

The geometric sequence aₙ = 10·0.7ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.5.

a₁ = 10r = 0.7ε = 0.5
n =
3.Medium
k|aₖ|¹ᐟᵏ123456789101112

Apply the root test to Σk=1∞ ((2k + 1)/(3k + 2))ᵏ. Find L = lim |ak|1/k.

ak = ((2k + 1)/(3k + 2))ᵏ
L =
4.Medium
xx₀ = −2??R

Find the radius of convergence R of the power series Σk=1∞ k!·(x + 2)ᵏ/kᵏ.

Σ k!·(x + 2)ᵏ/kᵏx₀ = −2
R =
5.Hard

f(x) = ln(1 + x) is approximated by its Maclaurin polynomial of degree 4 at x = 0.2. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 4! = 24 to find the largest possible error |R₄|.

f(x) = ln(1 + x)n = 4x = 0.2M = 24
|R₄| =
6.Hard

Find the coefficient c₄ of (x − 2)⁴ in the Taylor series of f(x) = e2x about a = 2.

f(x) = e2xa = 2k = 4
c₄ =

Answer key

  1. 1. n = 14
  2. 2. n = 10
  3. 3. L = 0.6667
  4. 4. R = 2.718
  5. 5. |R₄| = 6.4 × 10⁻⁵
  6. 6. c₄ = 36.4

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