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30.2 Conditions for Convergence of an Alternating Sequence

A sequence whose terms alternate in sign is called an alternating sequence, and such a sequence converges if two simple conditions hold:

1. Its terms decrease in magnitude: so we have | a j + 1 | ≤ | a j | .

2. The terms converge to 0.

The second of these conditions is necessary for convergence; the first is not.

Why are these conditions sufficient for convergence?

We can group successive pairs of terms together either from the start or after the first term. These lead us to the two following expressions

( a 1 + a 2 ) + ( a 3 + a 4 ) + ( a 5 + a 6 ) + ( a 7 + a 8 ) + ⋯

and

a 1 + ( a 2 + a 3 ) + ( a 4 + a 5 ) + ( a 6 + a 7 ) + ( a 8 + a 9 ) + ⋯

Notice that because the series is alternating in sign, the terms in parentheses are differences of the absolute values of successive terms, and by our first condition they all have the signs of their first terms.

Suppose, for example, that a 1 is positive. Then all the terms in the first expression are positive, and its partial sums will increase. On the other hand, all the terms in the second expression after the first will be negative, and its partial sums will decrease. Thus, the even partial sums of our series all be below the final sum and will increase while the odd partial sums will be above the final sum and decrease.

The value of all later partial sums must therefore at each stage lie sandwiched between any two successive sums.

By our second condition differences in successive partial sums (which are the terms in the series themselves) approach 0, which means that the values of the partial sums are constricted to intervals that approach 0 in size, which implies that the partial sums converge in the sense of Cauchy (their differences approach 0).

Every alternating series obeying the two conditions above with first term positive, has the property that the odd partial sums are decreasing and even ones increasing. This implies that averaging successive pairs of partial sums gives a better approximation to the total sum than the partial sums themselves do.

The alternating harmonic series has the wonderful property that after such averaging, the resulting even and odd averages are still increasing and decreasing, and this property is preserved if you average pairs of these, and then pairs of what you get, and so on. These facts can be used to find the value of this sequence to great accuracy, looking only at their first few terms.

You can average successive terms on a spread sheet by copying one instruction, and repeat doing so as indicated in the previous paragraph by copying it more. You will be surprised at how accurate you can determine its sum from say 25 terms, the smallest of which is .04.

Exercise 30.3 Use a spreadsheet to find the first 40 partial sums of the alternating harmonic sequence. Then take the sum of successive pairs and repeat 20 times. (This can be done with one instruction of the form d j = c j + c j − 1 2 , copied into columns d , e , f , g , … )
What do you find? How accurately can you determine the sum of this sequence from this data? From the first 20 partial sums alone?

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