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​12345678ε = 0.5

The geometric sequence aₙ = 10·0.6ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.5.

a₁ = 10r = 0.6ε = 0.5
n =
2.Easy
kak​1234567

Find the sum S7 of the first 7 terms of the geometric sequence 10, 20, 40, …

a₁ = 10r = 2n = 7
S7 =
3.Medium
AΣ 1/(2ᵏ + k)k = 1 … ∞BΣ 1/(5k + 5)k = 1 … ∞CΣ (7/4)ᵏk = 1 … ∞DΣ ln k / kk = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ 1/(2ᵏ + k)B: Σ 1/(5k + 5)C: Σ (7/4)ᵏD: Σ ln k / k
  1. 1A: Σ 1/(2ᵏ + k)
  2. 2B: Σ 1/(5k + 5)
  3. 3C: Σ (7/4)ᵏ
  4. 4D: Σ ln k / k
4.Medium
xx₀ = 2??R

The power series Σk=1∞ k²·5ᵏ·(x − 2)ᵏ converges on an interval around x₀ = 2. Find the left end x of that interval.

Σ k²·5ᵏ·(x − 2)ᵏx₀ = 2
x =
5.Hard

f(x) = sin x is approximated at x = 1.5 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) = sin xx = 1.5M = 1ε = 0.001
n =
6.Hard

f(x) = eˣ is approximated by its Maclaurin polynomial of degree 5 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 |R₅|.

f(x) = eˣn = 5x = 0.3M = 1.35
|R₅| =

Answer key

  1. 1. n = 7
  2. 2. S7 = 1270
  3. 3. A: Σ 1/(2ᵏ + k)
  4. 4. x = 1.8
  5. 5. n = 7
  6. 6. |R₅| = 1.367 × 10⁻⁶

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