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
kSn​12345678910S = ?

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

a = 2r = −2/3k₀ = 2
S =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=0∞ 8·(3/5)ᵏ.

a = 8r = 3/5k₀ = 0
S =
3.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 1/(k(k + 4)).

ak = 1/(k(k + 4))
S =
4.Medium
AΣ k³/2ᵏk = 1 … ∞BΣ (1 + 1/k)ᵏk = 1 … ∞CΣ 3/k⁴k = 1 … ∞DΣ (−1)ᵏ/√kk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ k³/2ᵏB: Σ (1 + 1/k)ᵏC: Σ 3/k⁴D: Σ (−1)ᵏ/√k
  1. 1A: Σ k³/2ᵏ
  2. 2B: Σ (1 + 1/k)ᵏ
  3. 3C: Σ 3/k⁴
  4. 4D: Σ (−1)ᵏ/√k
5.Hard

f(x) = cos x is approximated by its Maclaurin polynomial of degree 4 at x = 0.8. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₄|.

f(x) = cos xn = 4x = 0.8M = 1
|R₄| =
6.Hard

f(x) = eˣ 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 = 2, find the smallest degree n that guarantees an error below 0.001.

f(x) = eˣx = 0.5M = 2ε = 0.001
n =

Answer key

  1. 1. S = 0.5333
  2. 2. S = 20
  3. 3. S = 0.5208
  4. 4. B: Σ (1 + 1/k)ᵏ
  5. 5. |R₄| = 0.002731
  6. 6. n = 4

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