MathCS.org - Real Analysis: Examples 4.2.20(a):

Examples 4.2.20(a):

The Alternating Harmonic Series: The series

is called the Alternating Harmonic series. It converges but not absolutely, i.e. it converges conditionally.

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Proof:

There are many proofs of this fact. For example. the series of absolute values is a p-series with p = 1, and diverges by the p-series test. The original series converges, because it is an alternating series, and the alternating series test applies easily. However, here is a more elementary proof of the convergence of the alternating harmonic series.

We already know that the series of absolute values does not converge by a previous example. Hence, the series does not converge absolutely. As for regular convergence, consider the following two partial sums:

We have that

S _2n+2 - S _2n = 1 / (2n+1) - 1 / (2n+2) > 0
S _2n+3 - S _2n+1 = - 1 / (2n+2) + 1/ (2n+3) < 0

which means for the two subsequences

{ S _2n } is monotone increasing
{ S _2n+1 } is monotone decreasing

For each sequence we can combine pairs to see that

S _2n 1 for all n
S _2n+1 0 for all n

Hence, both subsequences are monotone and bounded, and must therefore be convergent. Define their limits as

lim S _2n = L and lim S _2n+1 = M

Then

| M - L | = | lim (S _2n+1 - S _2n) | = 1 / (2n+1)

which converges to zero. Therefore, M = L, i.e. both subsequences converge to the same limit. But this common limit is the same as the limit of the full sequence, because: given any > 0 we have

there exists an integer N such that | L - S _2n | < if n > N
there exists an integer M such that | L - S _2n+1 | < if n > M

Now set K = max(N, M). Then, for the above > 0 we have

| L - S _n | <

for n > K, because n is either even or odd. Hence, the alternating harmonic series converges conditionally.

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Examples 4.2.20(a):

Proof: