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​13579111315ε = 0.01

The geometric sequence aₙ = 10·0.9ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.01.

a₁ = 10r = 0.9ε = 0.01
n =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=2∞ 12·(−2/3)ᵏ.

a = 12r = −2/3k₀ = 2
S =
3.Medium
xx₀ = 3??R

The power series Σk=1∞ k²·3ᵏ·(x − 3)ᵏ converges on an interval around x₀ = 3. Find the left end x of that interval.

Σ k²·3ᵏ·(x − 3)ᵏx₀ = 3
x =
4.Medium
xx₀ = 3??R

The power series Σk=1∞ k!·(x − 3)ᵏ/kᵏ converges on an interval around x₀ = 3. Find the right end x of that interval.

Σ k!·(x − 3)ᵏ/kᵏx₀ = 3
x =
5.Hard
xyfP₃2.5

Use the Maclaurin polynomial of degree 3 of f(x) = sin(0.5x) to approximate f(2.5). Give P₃(2.5).

f(x) = sin(0.5x)n = 3x = 2.5
P₃(2.5) =
6.Hard

f(x) = sin 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 1 × 10⁻⁵.

f(x) = sin xx = 2M = 1ε = 1 × 10⁻⁵
n =

Answer key

  1. 1. n = 67
  2. 2. S = 3.2
  3. 3. x = 2.667
  4. 4. x = 5.718
  5. 5. P₃(2.5) = 0.9245
  6. 6. n = 11

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