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

Find the sum S16 of the first 16 terms of the arithmetic sequence −9, −13, −17, −21, …

a₁ = −9d = −4n = 16
S16 =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=1∞ 12·(2/5)ᵏ.

a = 12r = 2/5k₀ = 1
S =
3.Medium
AΣ (7/4)ᵏk = 1 … ∞BΣ (1 + 1/k)ᵏk = 1 … ∞CΣ k!/3ᵏk = 1 … ∞DΣ (−1)ᵏ/kk = 1 … ∞

Exactly one of the four series converges. Click it.

A: Σ (7/4)ᵏB: Σ (1 + 1/k)ᵏC: Σ k!/3ᵏD: Σ (−1)ᵏ/k
  1. 1A: Σ (7/4)ᵏ
  2. 2B: Σ (1 + 1/k)ᵏ
  3. 3C: Σ k!/3ᵏ
  4. 4D: Σ (−1)ᵏ/k
4.Medium
xx₀ = 3??R

The power series Σk=1∞ k!·(x − 3)ᵏ/kᵏ converges on an interval around x₀ = 3. Find the left end x of that interval.

Σ k!·(x − 3)ᵏ/kᵏx₀ = 3
x =
5.Hard
xyfP₃0.25

Use the Maclaurin polynomial of degree 3 of f(x) = 1/(1 − 2x) to approximate f(0.25). Give P₃(0.25).

f(x) = 1/(1 − 2x)n = 3x = 0.25
P₃(0.25) =
6.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 × 10⁻⁵.

f(x) = cos xx = 1M = 1ε = 1 × 10⁻⁵
n =

Answer key

  1. 1. S16 = −624
  2. 2. S = 8
  3. 3. D: Σ (−1)ᵏ/k
  4. 4. x = 0.2817
  5. 5. P₃(0.25) = 1.875
  6. 6. n = 8

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