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​123456789ε = 0.5

The geometric sequence aₙ = 2·0.8ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.5.

a₁ = 2r = 0.8ε = 0.5
n =
2.Easy
kak​123456789101112

Find the sum S28 of the first 28 terms of the arithmetic sequence 14, 16, 18, 20, …

a₁ = 14d = 2n = 28
S28 =
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₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ kᵏ/(3ᵏ·k!). Find L = lim |ak+1/ak|.

ak = kᵏ/(3ᵏ·k!)
L =
5.Hard

f(x) = eˣ is approximated at x = 1 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 3, find the smallest degree n that guarantees an error below 0.0001.

f(x) = eˣx = 1M = 3ε = 0.0001
n =
6.Hard

f(x) = cos x is approximated by its Maclaurin polynomial of degree 5 at x = 0.5. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₅|.

f(x) = cos xn = 5x = 0.5M = 1
|R₅| =

Answer key

  1. 1. n = 8
  2. 2. S28 = 1148
  3. 3. L = 0.3679
  4. 4. L = 0.9061
  5. 5. n = 7
  6. 6. |R₅| = 2.17 × 10⁻⁵

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