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Some Sum!

Find approximate values of the following partial sums, involving the alternating sums/differences of the reciprocals of the odd numbers:

  • 4(1)
  • 4(1 - 1/3)
  • 4(1 - 1/3 + 1/5)
  • 4(1 - 1/3 + 1/5 - 1/7)
  • 4(1 - 1/3 + 1/5 - 1/7 + 1/9)
  • 4(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11)
  • 4(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13)
  • 4(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15)
  • 4(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 + 1/17)
  • 4(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 + 1/17 - 1/19)
  • 4(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 + 1/17 - 1/19 + 1/21)
Question 1: What value is this series (and sequence of its partial sums) approaching?

Question 2: How could you prove it?

 


Hint: Think...what special day is involved!

 


Solution Commentary: This interesting series, known as the simplest series for approximating pi, is the Leibniz Series, discovered by Gottfried Wilhelm Leibniz in 1674.

For those who have taken calculus, Leibniz created this series by expanding the arctangent function using Maclaurin's series:

arctan(x) = x - x3/3 + x5/5 - x7/7 + x9 ... + (-1)n-1x2n-1/(2n-1) +....

which convereges for all x in the closed interval [-1,1]. If you let x = 1 (in radians, tan[pi/4]=1) you get the desired series expansion for pi/4, which leads to the series shown.