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​12345678910

Find the sum S10 of the first 10 terms of the geometric sequence 4, −12, 36, …

a₁ = 4r = −3n = 10
S10 =
2.Easy
kak​13579111315ε = 0.01

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 0.01 m?

a₁ = 8r = 0.75ε = 0.01
n =
3.Medium
xx₀ = 1??R

The power series Σk=1∞ 3ᵏ·(x − 1)ᵏ/(k + 1) converges on an interval around x₀ = 1. Find the right end x of that interval.

Σ 3ᵏ·(x − 1)ᵏ/(k + 1)x₀ = 1
x =
4.Medium
AΣ 5/k³k = 1 … ∞BΣ (2/5)ᵏk = 1 … ∞CΣ 3ᵏ/k!k = 1 … ∞DΣ 1/√kk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 5/k³B: Σ (2/5)ᵏC: Σ 3ᵏ/k!D: Σ 1/√k
  1. 1A: Σ 5/k³
  2. 2B: Σ (2/5)ᵏ
  3. 3C: Σ 3ᵏ/k!
  4. 4D: Σ 1/√k
5.Hard

f(x) = cos x is approximated at x = 1 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 0.0001.

f(x) = cos xx = 1M = 1ε = 0.0001
n =
6.Hard

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

f(x) = ln xa = 3k = 3
c₃ =

Answer key

  1. 1. S10 = −59048
  2. 2. n = 25
  3. 3. x = 1.333
  4. 4. D: Σ 1/√k
  5. 5. n = 7
  6. 6. c₃ = 0.01235

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