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 S35 of the first 35 terms of the arithmetic sequence 6, 9, 12, 15, …

a₁ = 6d = 3n = 35
S35 =
2.Easy
kSn​12345678910S = ?

Find the sum S of the infinite series Σk=0∞ 10·(−3/4)ᵏ.

a = 10r = −3/4k₀ = 0
S =
3.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 =
4.Medium
xx₀ = 0??R

Find the radius of convergence R of the power series Σk=1∞ 16ᵏx²ᵏ/k.

Σ 16ᵏx²ᵏ/kx₀ = 0
R =
5.Hard

f(x) = sin x 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 = 1, find the smallest degree n that guarantees an error below 0.0005.

f(x) = sin xx = 0.5M = 1ε = 0.0005
n =
6.Hard

f(x) = eˣ is approximated by its Maclaurin polynomial of degree 4 at x = 0.5. Use the Lagrange error bound M·|x|ⁿ⁺¹/(n + 1)! with M = e0.5 = 1.649 to find the largest possible error |R₄|.

f(x) = eˣn = 4x = 0.5M = 1.649
|R₄| =

Answer key

  1. 1. S35 = 1995
  2. 2. S = 5.714
  3. 3. L = 0.5437
  4. 4. R = 0.25
  5. 5. n = 4
  6. 6. |R₄| = 0.0004294

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