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
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=0∞ 18·(−1/4)ᵏ.

a = 18r = −1/4k₀ = 0
S =
2.Easy
kak​12345ε = 0.5

A ball is dropped and its first rebound reaches 3 m. Every next rebound reaches 50% of the previous one. Which rebound n is the first one lower than 0.5 m?

a₁ = 3r = 0.5ε = 0.5
n =
3.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 4/(k(k + 1)).

ak = 4/(k(k + 1))
S =
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ (k!)²·5ᵏ/(2k)!. Find L = lim |ak+1/ak|.

ak = (k!)²·5ᵏ/(2k)!
L =
5.Hard

f(x) = sin x is approximated by its Maclaurin polynomial of degree 4 at x = 0.8. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₄|.

f(x) = sin xn = 4x = 0.8M = 1
|R₄| =
6.Hard

Find the coefficient c₄ of (x + 1)⁴ in the Taylor series of f(x) = e3x about a = −1.

f(x) = e3xa = −1k = 4
c₄ =

Answer key

  1. 1. S = 14.4
  2. 2. n = 4
  3. 3. S = 4
  4. 4. L = 1.25
  5. 5. |R₄| = 0.002731
  6. 6. c₄ = 0.168

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