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​123456

Find the sum S6 of the first 6 terms of the geometric sequence 0.5, −0.25, 0.125, …

a₁ = 0.5r = −0.5n = 6
S6 =
2.Easy
kak​123456789101112

Find the term a32 of the arithmetic sequence −9, −3, 3, 9, …

a₁ = −9d = 6n = 32
a32 =
3.Medium
AΣ (4/5)ᵏk = 1 … ∞BΣ (−1)ᵏ/kk = 1 … ∞CΣ (5/3)ᵏk = 1 … ∞DΣ 2ᵏ/k!k = 1 … ∞

Exactly one of the four series diverges. Click it.

A: Σ (4/5)ᵏB: Σ (−1)ᵏ/kC: Σ (5/3)ᵏD: Σ 2ᵏ/k!
  1. 1A: Σ (4/5)ᵏ
  2. 2B: Σ (−1)ᵏ/k
  3. 3C: Σ (5/3)ᵏ
  4. 4D: Σ 2ᵏ/k!
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ (k!)²·8ᵏ/(2k)!. Find L = lim |ak+1/ak|.

ak = (k!)²·8ᵏ/(2k)!
L =
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 0.0005.

f(x) = eˣx = 2M = 8ε = 0.0005
n =
6.Hard

Find the coefficient c₂ of (x − 16)² in the Taylor series of f(x) = √x about a = 16.

f(x) = √xa = 16k = 2
c₂ =

Answer key

  1. 1. S6 = 0.3281
  2. 2. a32 = 177
  3. 3. C: Σ (5/3)ᵏ
  4. 4. L = 2
  5. 5. n = 10
  6. 6. c₂ = −0.001953

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