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​123456789101112

The arithmetic sequence 4, 1, −2, −5, … ends with the term −113. How many terms n does it have?

a₁ = 4d = −3aₙ = −113
n =
2.Easy
kak​123456789101112

Find the term a39 of the arithmetic sequence −10, −7.5, −5, −2.5, …

a₁ = −10d = 2.5n = 39
a39 =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ (k!)²·3ᵏ/(2k)!. Find L = lim |ak+1/ak|.

ak = (k!)²·3ᵏ/(2k)!
L =
4.Medium
AΣ (k + 3)/k²k = 1 … ∞BΣ (1/3)ᵏk = 1 … ∞CΣ 3/kk = 1 … ∞DΣ k/(k + 7)k = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ (k + 3)/k²B: Σ (1/3)ᵏC: Σ 3/kD: Σ k/(k + 7)
  1. 1A: Σ (k + 3)/k²
  2. 2B: Σ (1/3)ᵏ
  3. 3C: Σ 3/k
  4. 4D: Σ k/(k + 7)
5.Hard

Find the coefficient c₂ of (x − 1)² in the Taylor series of f(x) = e2x about a = 1.

f(x) = e2xa = 1k = 2
c₂ =
6.Hard

f(x) = cos x is approximated by its Maclaurin polynomial of degree 3 at x = 0.5. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = 1 to find the largest possible error |R₃|.

f(x) = cos xn = 3x = 0.5M = 1
|R₃| =

Answer key

  1. 1. n = 40
  2. 2. a39 = 85
  3. 3. L = 0.75
  4. 4. B: Σ (1/3)ᵏ
  5. 5. c₂ = 14.78
  6. 6. |R₃| = 0.002604

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