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=1∞ 8·(−2/3)ᵏ.

a = 8r = −2/3k₀ = 1
S =
2.Easy
kak​13579111315ε = 0.01

The geometric sequence aₙ = 8·0.75ⁿ⁻¹ decreases. Find the first index n for which aₙ < 0.01.

a₁ = 8r = 0.75ε = 0.01
n =
3.Medium
AΣ 3ᵏ/k!k = 1 … ∞BΣ (2/5)ᵏk = 1 … ∞CΣ (−1)ᵏ/√kk = 1 … ∞DΣ (3k + 1)/(k + 2)k = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 3ᵏ/k!B: Σ (2/5)ᵏC: Σ (−1)ᵏ/√kD: Σ (3k + 1)/(k + 2)
  1. 1A: Σ 3ᵏ/k!
  2. 2B: Σ (2/5)ᵏ
  3. 3C: Σ (−1)ᵏ/√k
  4. 4D: Σ (3k + 1)/(k + 2)
4.Medium
k|aₖ|¹ᐟᵏ123456789101112

Apply the root test to Σk=1∞ ((5k + 3)/(2k + 1))ᵏ. Find L = lim |ak|1/k.

ak = ((5k + 3)/(2k + 1))ᵏ
L =
5.Hard

Recognise the series and find its sum: S = Σk=0∞ (−1)ᵏ·22k+1/(2k + 1)!.

ak = (−1)ᵏ·22k+1/(2k + 1)!
S =
6.Hard

Find the coefficient c₃ of (x − 4)³ in the Taylor series of f(x) = ln x about a = 4.

f(x) = ln xa = 4k = 3
c₃ =

Answer key

  1. 1. S = −3.2
  2. 2. n = 25
  3. 3. D: Σ (3k + 1)/(k + 2)
  4. 4. L = 2.5
  5. 5. S = 0.9093
  6. 6. c₃ = 0.005208

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