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=1∞ 18·(1/3)ᵏ.

a = 18r = 1/3k₀ = 1
S =
2.Easy
kak​123456789

Find the sum S9 of the first 9 terms of the geometric sequence 1, 0.8, 0.64, …

a₁ = 1r = 0.8n = 9
S9 =
3.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 =
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ k²·3ᵏ/4ᵏ. Find L = lim |ak+1/ak|.

ak = k²·3ᵏ/4ᵏ
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 0.0005.

f(x) = cos xx = 2M = 1ε = 0.0005
n =
6.Hard

f(x) = ln(1 + x) is approximated by its Maclaurin polynomial of degree 4 at x = 0.5. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 4! = 24 to find the largest possible error |R₄|.

f(x) = ln(1 + x)n = 4x = 0.5M = 24
|R₄| =

Answer key

  1. 1. S = 9
  2. 2. S9 = 4.329
  3. 3. L = 0.9061
  4. 4. L = 0.75
  5. 5. n = 9
  6. 6. |R₄| = 0.00625

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