10.5.3 The Alternating Harmonic Series

The presence of alternating signs in the alternating harmonic series produces a quite different result from what we have just seen with the harmonic series. We start by introducing notation for the partial sums of the series:

and, in general,

The question of whether or not the infinite series

converges to a limiting value `S` is the same question as whether

We investigate this question in the next two activities.

Activity 3

1. Draw a horizontal line the width of a piece of paper. Near the left end, mark a point `0` and near the right end mark a point `2`. Carefully measure the midpoint, and label it `1`. Now calculate and place on your number line the numbers `s_1, s_2, s_3, s_4, s_5, s_6, s_7, ` and `s_8`.

2. Use your graphing tool to graph, in the same window, both the terms

     b n = ( - 1 ) n + 1 1 n

   and the partial sums

     s n = 1 - 1 2 + 1 3 - 1 4 + ⋅⋅⋅ + ( - 1 ) n + 1 1 n

   for `n = 1,2,...,20`.

3. Show that

     s 1 > s 3 > s 5 > ⋅⋅⋅

   for odd-numbered partial sums, and

     s 2 < s 4 < s 6 < ⋅⋅⋅

   for even-numbered ones.

4. Describe the general pattern of the sums `s_1,s_2,...,s_n` for an arbitrary positive integer `n`.

5. Show that

     lim n → ∞     | s n - s n + 1 | = 0.

6. Use these observations to show that the alternating harmonic series converges, i.e., that there really is a limiting value `S`.

7. Explain why `S` is smaller than every odd-numbered partial sum and larger than every even-numbered partial sum.

8. The error after summing `n` terms is `text[|] S - s_n text[|]`. Explain why this error is smaller than the absolute value of the `text[(] n+1 text[)]`th term, that is, `1 text[/] text[(] n + 1 text[)] .`

Comment on Activity 3

Checkpoint 1