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​123456789101112

The arithmetic sequence −8, −5, −2, 1, … ends with the term 43. How many terms n does it have?

a₁ = −8d = 3aₙ = 43
n =
2.Easy
kak​1234567

Find the sum S7 of the first 7 terms of the geometric sequence 10, 5, 2.5, …

a₁ = 10r = 0.5n = 7
S7 =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ (k!)²·8ᵏ/(2k)!. Find L = lim |ak+1/ak|.

ak = (k!)²·8ᵏ/(2k)!
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 right end x of that interval.

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

f(x) = sin x is approximated by its Maclaurin polynomial of degree 2 at x = −0.5. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₂|.

f(x) = sin xn = 2x = −0.5M = 1
|R₂| =
6.Hard

Find the coefficient c₃ of (x − 2)³ in the Taylor series of f(x) = e2x about a = 2.

f(x) = e2xa = 2k = 3
c₃ =

Answer key

  1. 1. n = 18
  2. 2. S7 = 19.84
  3. 3. L = 2
  4. 4. x = 1.25
  5. 5. |R₂| = 0.02083
  6. 6. c₃ = 72.8

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