Alternating series

Definition

An alternating series is a series containing terms with alternating signs. Here are two examples:

 ∞
---     n-1
\   (-1)          1   1   1
/   ------- = 1 - - + - - - + ...
---    n          2   3   4
n=1

 ∞
---     n
\   (-1) n     1   2   3   4
/   ------ = - - + - - - + - + ...
---  n + 1     2   3   4   5
n=1

Looking at our examples we notice that the n-th term of an alternating series can be described in two ways:

a_n=(-1)^n-1b_n     or     a_n=(-1)ⁿb_n

Where b_n is a positive number. More generally, b_n = |a_n|.

Alternating Series Test

As is with all series mathematics, the fundamental question we must ask is, "Does this series converge?". With alternating series, mathematicians have come up with a special test to handle the case, dubiously named the Alternating Series Test.

If the alternating series

 ∞
---     n-1
\   (-1)   b  = b  - b  + b  - b  + ...   b  > 0
/           n    1    2    3    4          n
---
n=1

satisfies

1.) b_n+1 ≤ b_n for all n
2.) The limit of b_n as n approaches infinity equals zero.

Then the series converges.

Before giving a proof for this series test, I want to draw a picture that almost makes the proof obsolete. First we plot s₁ = b₁ on a number line, then find s₂ by subtracting b₂, etc, etc. We find that since b_n approaches zero, the partial sums oscillate back and forth until the series converges to a sum. The even partial sums s_2n are increasing and the odd decrease, and therefore it is plausible that both converge to some number s.

                         +b(1)
    |----------------------------------------->|
    |                       -b(2)              |
    |         |<-------------------------------|
    |         |       +b(3)                    |
    |         |------------------------>|      |
    |         |             -b(4)       |      |
    |         |       |<----------------|      |
    |         |       |                 |      |
----|---------|-------|---------|-------|------|---------
    0         s       s         s        s      s
               2       4                 3      1

s₂ = b₁ - b₂ ≥ 0           since b₂ ≤ b₁

s₄ = s₂ + (b₃ - b₄) ≥ s₂    since b₄ ≤ b₃

In general,         s_2n = s_2n-2 + (b_2n-1 - b_2n) ≥ s_2n-2       since b_2n ≤ b_2n-1

Thus      0 ≤ s₂ ≤ s₄ ≤ s₆ ≤ ... ≤ s_2n ≤ ...

But we can also write:
s_2n = b₁ - (b₂ - b₃) - (s₄ - s₅) - ... - (b_2n-2 - b_2n-1) - b_2n

Since every term in parentheses is positive, s_2n ≤ b₁ for all n. Therefore the sequence {s_2n} or the even partial sums is increasing and bounded above by b₁. Thus, by the Monotonic Sequence Theorem which says "Every bounded, monotonic sequence is convergent," our series must have a sum.

Well known Alternating Series

S(x) = sin(x)

        ∞
       ---           (2n+1)
S(x) = \       n    x
       /   (-1)  ------
       ---       (2n+1)!
       n=0

C(x) = cos(x)

        ∞
       ---           (2n)
C(x) = \       n    x
       /   (-1)   -----
       ---        (2n)!
       n=0