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 5 m. Every next rebound reaches 85% of the previous one. Which rebound n is the first one lower than 0.5 m?

a₁ = 5r = 0.85ε = 0.5
n =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=1∞ 6·(−1/4)ᵏ.

a = 6r = −1/4k₀ = 1
S =
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
k|aₖ₊₁/aₖ|123456789101112

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

ak = (k!)²·1ᵏ/(2k)!
L =
5.Hard

Find the coefficient c₂ of (x − 4)² in the Taylor series of f(x) = ln x about a = 4.

f(x) = ln xa = 4k = 2
c₂ =
6.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 0.001.

f(x) = cos xx = 1M = 1ε = 0.001
n =

Answer key

  1. 1. n = 16
  2. 2. S = −1.2
  3. 3. L = 1.104
  4. 4. L = 0.25
  5. 5. c₂ = −0.03125
  6. 6. n = 6

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