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, 1.5, 4.5, …

a₁ = 0.5r = 3n = 6
S6 =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=1∞ 3·(2/5)ᵏ.

a = 3r = 2/5k₀ = 1
S =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

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

ak = (k!)²·2ᵏ/(2k)!
L =
4.Medium
xx₀ = 1??R

The power series Σk=1∞ (x − 1)ᵏ/(k²·2ᵏ) converges on an interval around x₀ = 1. Find the left end x of that interval.

Σ (x − 1)ᵏ/(k²·2ᵏ)x₀ = 1
x =
5.Hard

f(x) = cos x is approximated at x = 1 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 = 1M = 1ε = 1 × 10⁻⁶
n =
6.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 0.0001.

f(x) = eˣx = 2M = 8ε = 0.0001
n =

Answer key

  1. 1. S6 = 182
  2. 2. S = 2
  3. 3. L = 0.5
  4. 4. x = −1
  5. 5. n = 9
  6. 6. n = 11

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