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​123456

Find the sum S6 of the first 6 terms of the geometric sequence 5, −15, 45, …

a₁ = 5r = −3n = 6
S6 =
2.Easy
kak​12345678910

Find the sum S10 of the first 10 terms of the geometric sequence 5, 2.5, 1.25, …

a₁ = 5r = 0.5n = 10
S10 =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ k!/kᵏ. Find L = lim |ak+1/ak|.

ak = k!/kᵏ
L =
4.Medium
k|aₖ|¹ᐟᵏ123456789101112

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

ak = ((1k + 6)/(6k + 6))ᵏ
L =
5.Hard

f(x) = cos x is approximated at x = 2 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 = 2M = 1ε = 0.001
n =
6.Hard

f(x) = eˣ is approximated by its Maclaurin polynomial of degree 2 at x = 1.5. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = e1.5 = 4.482 to find the largest possible error |R₂|.

f(x) = eˣn = 2x = 1.5M = 4.482
|R₂| =

Answer key

  1. 1. S6 = −910
  2. 2. S10 = 9.99
  3. 3. L = 0.3679
  4. 4. L = 0.1667
  5. 5. n = 9
  6. 6. |R₂| = 2.521

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