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∞ 9·(1/3)ᵏ.

a = 9r = 1/3k₀ = 1
S =
2.Easy
kak​13579111315ε = 0.1

A ball is dropped and its first rebound reaches 8 m. Every next rebound reaches 75% of the previous one. Which rebound n is the first one lower than 0.1 m?

a₁ = 8r = 0.75ε = 0.1
n =
3.Medium
AΣ 2/k⁴k = 1 … ∞BΣ (3/4)ᵏk = 1 … ∞CΣ ln k / kk = 1 … ∞DΣ (−1)ᵏ/√kk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 2/k⁴B: Σ (3/4)ᵏC: Σ ln k / kD: Σ (−1)ᵏ/√k
  1. 1A: Σ 2/k⁴
  2. 2B: Σ (3/4)ᵏ
  3. 3C: Σ ln k / k
  4. 4D: Σ (−1)ᵏ/√k
4.Medium
k|aₖ₊₁/aₖ|123456789101112

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

ak = kᵏ/(5ᵏ·k!)
L =
5.Hard

Find the coefficient c₂ of (x − 1)² in the Taylor series of f(x) = e−x about a = 1.

f(x) = e−xa = 1k = 2
c₂ =
6.Hard

Recognise the series and find its sum: S = Σk=0∞ (−1)ᵏ·0.82k/(2k)!.

ak = (−1)ᵏ·0.82k/(2k)!
S =

Answer key

  1. 1. S = 4.5
  2. 2. n = 17
  3. 3. C: Σ ln k / k
  4. 4. L = 0.5437
  5. 5. c₂ = 0.1839
  6. 6. S = 0.6967

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