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 = 5·0.75ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.01.

a₁ = 5r = 0.75ε = 0.01
n =
2.Easy
kak​123456789101112

Find the term a40 of the arithmetic sequence −11, −7, −3, 1, …

a₁ = −11d = 4n = 40
a40 =
3.Medium
AΣ 1/√kk = 1 … ∞BΣ (1 + 1/k)ᵏk = 1 … ∞CΣ ln k / kk = 1 … ∞DΣ (−1)ᵏ/√kk = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ 1/√kB: Σ (1 + 1/k)ᵏC: Σ ln k / kD: Σ (−1)ᵏ/√k
  1. 1A: Σ 1/√k
  2. 2B: Σ (1 + 1/k)ᵏ
  3. 3C: Σ ln k / k
  4. 4D: Σ (−1)ᵏ/√k
4.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 2/(k(k + 4)).

ak = 2/(k(k + 4))
S =
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 0.001.

f(x) = cos xx = 2M = 1ε = 0.001
n =
6.Hard
xyfP₃−1.5

Use the Maclaurin polynomial of degree 3 of f(x) = e−x to approximate f(−1.5). Give P₃(−1.5).

f(x) = e−xn = 3x = −1.5
P₃(−1.5) =

Answer key

  1. 1. n = 23
  2. 2. a40 = 145
  3. 3. D: Σ (−1)ᵏ/√k
  4. 4. S = 1.042
  5. 5. n = 9
  6. 6. P₃(−1.5) = 4.188

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