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 = 0.01

A ball is dropped and its first rebound reaches 20 m. Every next rebound reaches 60% of the previous one. Which rebound n is the first one lower than 0.01 m?

a₁ = 20r = 0.6ε = 0.01
n =
2.Easy
kak​135791113ε = 0.05

A ball is dropped and its first rebound reaches 3 m. Every next rebound reaches 70% of the previous one. Which rebound n is the first one lower than 0.05 m?

a₁ = 3r = 0.7ε = 0.05
n =
3.Medium
AΣ (−1)ᵏ/√kk = 1 … ∞BΣ ln k / kk = 1 … ∞CΣ (1 + 1/k)ᵏk = 1 … ∞DΣ (5/4)ᵏk = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ (−1)ᵏ/√kB: Σ ln k / kC: Σ (1 + 1/k)ᵏD: Σ (5/4)ᵏ
  1. 1A: Σ (−1)ᵏ/√k
  2. 2B: Σ ln k / k
  3. 3C: Σ (1 + 1/k)ᵏ
  4. 4D: Σ (5/4)ᵏ
4.Medium
kSn​12345678910S = ?

Find the sum S = Σk=1∞ 1/(k(k + 1)).

ak = 1/(k(k + 1))
S =
5.Hard

Find the coefficient c₃ of (x − 1)³ in the Taylor series of f(x) = e−x about a = 1.

f(x) = e−xa = 1k = 3
c₃ =
6.Hard

f(x) = sin x is approximated at x = 2 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 0.0005.

f(x) = sin xx = 2M = 1ε = 0.0005
n =

Answer key

  1. 1. n = 16
  2. 2. n = 13
  3. 3. A: Σ (−1)ᵏ/√k
  4. 4. S = 1
  5. 5. c₃ = −0.06131
  6. 6. n = 9

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