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∞ 8·(−1/4)ᵏ.

a = 8r = −1/4k₀ = 0
S =
2.Easy
kak​12345678ε = 0.5

A ball is dropped and its first rebound reaches 20 m. Every next rebound reaches 50% of the previous one. Which rebound n is the first one lower than 0.5 m?

a₁ = 20r = 0.5ε = 0.5
n =
3.Medium
k|aₖ₊₁/aₖ|123456789101112

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

ak = 2ᵏ·k!/kᵏ
L =
4.Medium
k|aₖ₊₁/aₖ|123456789101112

Apply the ratio test to Σk=1∞ kᵏ/(5ᵏ·k!). Find L = lim |ak+1/ak|.

ak = kᵏ/(5ᵏ·k!)
L =
5.Hard

f(x) = eˣ is approximated by its Maclaurin polynomial of degree 4 at x = 1.5. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = e1.5 = 4.482 to find the largest possible error |R₄|.

f(x) = eˣn = 4x = 1.5M = 4.482
|R₄| =
6.Hard

f(x) = eˣ is approximated at x = 1 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 3, find the smallest degree n that guarantees an error below 1 × 10⁻⁵.

f(x) = eˣx = 1M = 3ε = 1 × 10⁻⁵
n =

Answer key

  1. 1. S = 6.4
  2. 2. n = 7
  3. 3. L = 0.7358
  4. 4. L = 0.5437
  5. 5. |R₄| = 0.2836
  6. 6. n = 8

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