**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-1}*b*_{n} or *a*_{n}=(-1)^{n}*b*_{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 *s*_{1} = *b*_{1} on a number line, then find *s*_{2} by subtracting *b*_{2}, 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

**Proof**
*s*_{2} = *b*_{1} - *b*_{2} ≥ 0 since *b*_{2} ≤ *b*_{1}

*s*_{4} = *s*_{2} + (*b*_{3} - *b*_{4}) ≥ *s*_{2} since *b*_{4} ≤ *b*_{3}

In general,
*s*_{2n} = *s*_{2n-2} + (*b*_{2n-1} - *b*_{2n}) ≥ *s*_{2n-2} since *b*_{2n} ≤ *b*_{2n-1}

Thus 0 ≤ *s*_{2} ≤ *s*_{4} ≤ *s*_{6} ≤ ... ≤ *s*_{2n} ≤ ...

But we can also write:

*s*_{2n} = *b*_{1} - (*b*_{2} - *b*_{3}) - (*s*_{4} - *s*_{5}) - ... - (*b*_{2n-2} - *b*_{2n-1}) - *b*_{2n}

Since every term in parentheses is positive, *s*_{2n} ≤ *b*_{1} for all *n*. Therefore the sequence {*s*_{2n}} or the even partial sums is increasing and bounded above by *b*_{1}. 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