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

Find the term a23 of the arithmetic sequence −7, −5, −3, −1, …

a₁ = −7d = 2n = 23
a23 =
2.Easy
kak​123456789101112

The arithmetic sequence 2, 6, 10, 14, … ends with the term 142. How many terms n does it have?

a₁ = 2d = 4aₙ = 142
n =
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∞ 2ᵏ·k!/kᵏ. Find L = lim |ak+1/ak|.

ak = 2ᵏ·k!/kᵏ
L =
5.Hard

f(x) = eˣ is approximated at x = 2 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 8, find the smallest degree n that guarantees an error below 0.0001.

f(x) = eˣx = 2M = 8ε = 0.0001
n =
6.Hard

f(x) = ln(1 + x) is approximated by its Maclaurin polynomial of degree 5 at x = 0.2. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 5! = 120 to find the largest possible error |R₅|.

f(x) = ln(1 + x)n = 5x = 0.2M = 120
|R₅| =

Answer key

  1. 1. a23 = 37
  2. 2. n = 36
  3. 3. L = 1.104
  4. 4. L = 0.7358
  5. 5. n = 11
  6. 6. |R₅| = 1.067 × 10⁻⁵

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