Chapter 10
Polynomial and Series Representations of Functions





10.5 Series of Constants

10.5.4 The Leibniz Series

Here again is the formula for the arctangent series:

arctan ⁢   x = x - 1 3 x 3 + 1 5 x 5 - ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k x 2 k + 1 2 k + 1 , if `text[|] x text[|] < 1.`

The value `x = 1` is not in the presumed domain `text[|] x text[|] < 1` for this series. However, if we substitute this value into the series, we get another interesting series, which has a name.


Definition   The Leibniz series is the series of constants
1 - 1 3 + 1 5 - 1 7 + ⋅⋅⋅ + ( - 1 ) k 1 2 k + 1 + ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k 1 2 k + 1 .

The question of immediate interest is whether the Leibniz series actually converges to `text(arctan) 1 = pi//4`. If so, it produces an interesting formula for `pi`:

π = 4 [ 1 - 1 3 + 1 5 - 1 7 + ⋅⋅⋅ + ( - 1 ) k 1 2 k + 1 + ⋅⋅⋅ ] .

Activity 4

  1. Explain why the Leibniz series converges to something. (Hint: See Activity 3.)

  2. Use your computer algebra system to add up the first `100` partial sums. Are you ready to believe that the limiting value might be `pi//4`? Why or why not? Change `100` to `1000` and answer the same questions.

  3. Whatever the actual sum of the Leibniz series is, let's call it `S`, explain why `S` is smaller than every even-numbered partial sum and larger than every odd-numbered partial sum.

  4. Explain why the error after summing `n` terms (numbered `0` through `n - 1`) is smaller than the absolute value of the `text[(]n + 1text[)]`th term, that is, `1 text[/] text[(] 2n + 1 text[)]`.

Comment 4Comment on Activity 4

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