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​1234567

Find the sum S7 of the first 7 terms of the geometric sequence 2, 3, 4.5, …

a₁ = 2r = 1.5n = 7
S7 =
2.Easy
kak​12345678910ε = 0.1

A ball is dropped and its first rebound reaches 5 m. Every next rebound reaches 60% of the previous one. Which rebound n is the first one lower than 0.1 m?

a₁ = 5r = 0.6ε = 0.1
n =
3.Medium
AΣ 2/k⁴k = 1 … ∞BΣ (1/3)ᵏk = 1 … ∞CΣ 1/(3k + 5)k = 1 … ∞DΣ 1/(3ᵏ + k)k = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ 2/k⁴B: Σ (1/3)ᵏC: Σ 1/(3k + 5)D: Σ 1/(3ᵏ + k)
  1. 1A: Σ 2/k⁴
  2. 2B: Σ (1/3)ᵏ
  3. 3C: Σ 1/(3k + 5)
  4. 4D: Σ 1/(3ᵏ + k)
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ k·3ᵏ/5ᵏ. Find L = lim |ak+1/ak|.

ak = k·3ᵏ/5ᵏ
L =
5.Hard

f(x) = sin x is approximated at x = 1.5 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 = 1.5M = 1ε = 0.0005
n =
6.Hard
xyfP₃0.5

Use the Maclaurin polynomial of degree 3 of f(x) = ln(1 + x) to approximate f(0.5). Give P₃(0.5).

f(x) = ln(1 + x)n = 3x = 0.5
P₃(0.5) =

Answer key

  1. 1. S7 = 64.34
  2. 2. n = 9
  3. 3. C: Σ 1/(3k + 5)
  4. 4. L = 0.6
  5. 5. n = 8
  6. 6. P₃(0.5) = 0.4167

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