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 5 m. Every next rebound reaches 75% of the previous one. Which rebound n is the first one lower than 0.01 m?

a₁ = 5r = 0.75ε = 0.01
n =
2.Easy
kak​13579111315ε = 0.2

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 0.2 m?

a₁ = 2r = 0.85ε = 0.2
n =
3.Medium
k|aₖ|¹ᐟᵏ123456789101112

Apply the root test to Σk=1∞ ((5k + 4)/(6k + 1))ᵏ. Find L = lim |ak|1/k.

ak = ((5k + 4)/(6k + 1))ᵏ
L =
4.Medium
xx₀ = 1??R

The power series Σk=1∞ (2k)!·(x − 1)ᵏ/(k!)² converges on an interval around x₀ = 1. Find the left end x of that interval.

Σ (2k)!·(x − 1)ᵏ/(k!)²x₀ = 1
x =
5.Hard

Find the coefficient c₃ of (x − 1)³ in the Taylor series of f(x) = 1/x about a = 1.

f(x) = 1/xa = 1k = 3
c₃ =
6.Hard

f(x) = eˣ is approximated by its Maclaurin polynomial of degree 3 at x = 0.8. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = e0.8 = 2.226 to find the largest possible error |R₃|.

f(x) = eˣn = 3x = 0.8M = 2.226
|R₃| =

Answer key

  1. 1. n = 23
  2. 2. n = 16
  3. 3. L = 0.8333
  4. 4. x = 0.75
  5. 5. c₃ = −1
  6. 6. |R₃| = 0.03798

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