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=0∞ 9·(4/5)ᵏ.

a = 9r = 4/5k₀ = 0
S =
2.Easy
kak​13579111315ε = 0.01

A ball is dropped and its first rebound reaches 10 m. Every next rebound reaches 85% of the previous one. Which rebound n is the first one lower than 0.01 m?

a₁ = 10r = 0.85ε = 0.01
n =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

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

ak = (k!)²·1ᵏ/(2k)!
L =
4.Medium
AΣ (−1)ᵏ/kk = 1 … ∞BΣ k/2ᵏk = 1 … ∞CΣ 1/(k² + 4)k = 1 … ∞DΣ 2/kk = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ (−1)ᵏ/kB: Σ k/2ᵏC: Σ 1/(k² + 4)D: Σ 2/k
  1. 1A: Σ (−1)ᵏ/k
  2. 2B: Σ k/2ᵏ
  3. 3C: Σ 1/(k² + 4)
  4. 4D: Σ 2/k
5.Hard

f(x) = eˣ is approximated at x = 2 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 8, find the smallest degree n that guarantees an error below 0.001.

f(x) = eˣx = 2M = 8ε = 0.001
n =
6.Hard
xyfP₂2.5

Use the Maclaurin polynomial of degree 2 of f(x) = e0.5x to approximate f(2.5). Give P₂(2.5).

f(x) = e0.5xn = 2x = 2.5
P₂(2.5) =

Answer key

  1. 1. S = 45
  2. 2. n = 44
  3. 3. L = 0.25
  4. 4. D: Σ 2/k
  5. 5. n = 10
  6. 6. P₂(2.5) = 3.031

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