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∞ 6·(4/5)ᵏ.

a = 6r = 4/5k₀ = 0
S =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=0∞ 12·(1/2)ᵏ.

a = 12r = 1/2k₀ = 0
S =
3.Medium
k|aₖ|¹ᐟᵏ123456789101112

Apply the root test to Σk=1∞ ((3k + 5)/(5k + 5))ᵏ. Find L = lim |ak|1/k.

ak = ((3k + 5)/(5k + 5))ᵏ
L =
4.Medium
k|aₖ₊₁/aₖ|123456789101112

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

ak = k·2ᵏ/3ᵏ
L =
5.Hard

f(x) = cos 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 1 × 10⁻⁶.

f(x) = cos xx = 1.5M = 1ε = 1 × 10⁻⁶
n =
6.Hard

Recognise the series and find its sum: S = Σk=0∞ (−1)ᵏ/(2k)!.

ak = (−1)ᵏ/(2k)!
S =

Answer key

  1. 1. S = 30
  2. 2. S = 24
  3. 3. L = 0.6
  4. 4. L = 0.6667
  5. 5. n = 11
  6. 6. S = 0.5403

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