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​135791113ε = 0.05

The geometric sequence aₙ = 20·0.6ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.05.

a₁ = 20r = 0.6ε = 0.05
n =
2.Easy
kak​1234567

Find the sum S7 of the first 7 terms of the geometric sequence 5, 15, 45, …

a₁ = 5r = 3n = 7
S7 =
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
xx₀ = 3??R

Find the radius of convergence R of the power series Σk=1∞ (2k)!·(x − 3)ᵏ/(k!)².

Σ (2k)!·(x − 3)ᵏ/(k!)²x₀ = 3
R =
5.Hard

f(x) = eˣ is approximated by its Maclaurin polynomial of degree 3 at x = 0.3. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = e0.3 = 1.35 to find the largest possible error |R₃|.

f(x) = eˣn = 3x = 0.3M = 1.35
|R₃| =
6.Hard

f(x) = ln(1 + x) is approximated by its Maclaurin polynomial of degree 2 at x = 0.1. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 2! = 2 to find the largest possible error |R₂|.

f(x) = ln(1 + x)n = 2x = 0.1M = 2
|R₂| =

Answer key

  1. 1. n = 13
  2. 2. S7 = 5465
  3. 3. L = 0.3679
  4. 4. R = 0.25
  5. 5. |R₃| = 0.0004556
  6. 6. |R₂| = 0.0003333

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