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 = 0.1

A ball is dropped and its first rebound reaches 10 m. Every next rebound reaches 75% of the previous one. Which rebound n is the first one lower than 0.1 m?

a₁ = 10r = 0.75ε = 0.1
n =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=2∞ 10·(−3/4)ᵏ.

a = 10r = −3/4k₀ = 2
S =
3.Medium
AΣ k²/3ᵏk = 1 … ∞BΣ ln k / kk = 1 … ∞CΣ (1 + 1/k)ᵏk = 1 … ∞DΣ 2ᵏ/k³k = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ k²/3ᵏB: Σ ln k / kC: Σ (1 + 1/k)ᵏD: Σ 2ᵏ/k³
  1. 1A: Σ k²/3ᵏ
  2. 2B: Σ ln k / k
  3. 3C: Σ (1 + 1/k)ᵏ
  4. 4D: Σ 2ᵏ/k³
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ k³·2ᵏ/5ᵏ. Find L = lim |ak+1/ak|.

ak = k³·2ᵏ/5ᵏ
L =
5.Hard

f(x) = eˣ is approximated at x = 1 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 3, find the smallest degree n that guarantees an error below 0.0005.

f(x) = eˣx = 1M = 3ε = 0.0005
n =
6.Hard

Recognise the series and find its sum: S = Σk=1∞ k·0.5ᵏ.

ak = k·0.5ᵏ
S =

Answer key

  1. 1. n = 18
  2. 2. S = 3.214
  3. 3. A: Σ k²/3ᵏ
  4. 4. L = 0.4
  5. 5. n = 7
  6. 6. S = 2

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