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 S33 of the first 33 terms of the arithmetic sequence −6, −10, −14, −18, …

a₁ = −6d = −4n = 33
S33 =
2.Easy
kak​135791113ε = 0.01

A ball is dropped and its first rebound reaches 3 m. Every next rebound reaches 60% of the previous one. Which rebound n is the first one lower than 0.01 m?

a₁ = 3r = 0.6ε = 0.01
n =
3.Medium
k|aₖ|¹ᐟᵏ123456789101112

Apply the root test to Σk=1∞ ((5k + 5)/(2k + 4))ᵏ. Find L = lim |ak|1/k.

ak = ((5k + 5)/(2k + 4))ᵏ
L =
4.Medium
k|aₖ₊₁/aₖ|123456789101112

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

ak = 3ᵏ·k!/kᵏ
L =
5.Hard

f(x) = eˣ is approximated at x = 0.5 by its Maclaurin polynomial of degree n. With the Lagrange bound M·|x|ⁿ⁺¹/(n + 1)! and M = 2, find the smallest degree n that guarantees an error below 0.0005.

f(x) = eˣx = 0.5M = 2ε = 0.0005
n =
6.Hard
xyfP₂−0.5

Use the Maclaurin polynomial of degree 2 of f(x) = cos(2x) to approximate f(−0.5). Give P₂(−0.5).

f(x) = cos(2x)n = 2x = −0.5
P₂(−0.5) =

Answer key

  1. 1. S33 = −2310
  2. 2. n = 13
  3. 3. L = 2.5
  4. 4. L = 1.104
  5. 5. n = 5
  6. 6. P₂(−0.5) = 0.5

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