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​12345

Find the sum S5 of the first 5 terms of the geometric sequence 4, 3.2, 2.56, …

a₁ = 4r = 0.8n = 5
S5 =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=0∞ 6·(2/5)ᵏ.

a = 6r = 2/5k₀ = 0
S =
3.Medium
k|aₖ|¹ᐟᵏ123456789101112

Apply the root test to Σk=1∞ ((1k + 1)/(5k + 1))ᵏ. Find L = lim |ak|1/k.

ak = ((1k + 1)/(5k + 1))ᵏ
L =
4.Medium
AΣ 1/k²k = 1 … ∞BΣ (−1)ᵏ/kk = 1 … ∞CΣ ln k / kk = 1 … ∞DΣ (2/5)ᵏk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 1/k²B: Σ (−1)ᵏ/kC: Σ ln k / kD: Σ (2/5)ᵏ
  1. 1A: Σ 1/k²
  2. 2B: Σ (−1)ᵏ/k
  3. 3C: Σ ln k / k
  4. 4D: Σ (2/5)ᵏ
5.Hard

f(x) = eˣ 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 = 1 to find the largest possible error |R₄|.

f(x) = eˣn = 4x = −0.5M = 1
|R₄| =
6.Hard

f(x) = cos x is approximated at x = 0.5 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 1 × 10⁻⁶.

f(x) = cos xx = 0.5M = 1ε = 1 × 10⁻⁶
n =

Answer key

  1. 1. S5 = 13.45
  2. 2. S = 10
  3. 3. L = 0.2
  4. 4. C: Σ ln k / k
  5. 5. |R₄| = 0.0002604
  6. 6. n = 7

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