In the previous post on the harmonic series, I never properly explained how the value of a series is determined. After all, if a sum contains an infinite number of terms, one can’t compute the sum by just adding all the terms together; that would take forever. Instead, the value of a series is defined as the limit of the sum of the first terms in the series as approaches infinity. For every positive integer , the sum of the first terms in the series is called its th partial sum, so another way of stating this definition is that the value of the series is the value to which its sequence of partial sums converges.
The thing about using this definition is that it makes it clear that infinite sums cannot necessarily be manipulated in the same way as finite ones. In the previous post we freely manipulated infinite sums without bothering to check whether these manipulations were justified. I’ll go through each of the manipulations in turn and prove them in a more rigorous manner.
The first thing we did in proving that the harmonic series diverges was to construct the new series
and note that since every term in this series is less than or equal to the corresponding term in the harmonic series, its value is infinite only if the value of the harmonic series is infinite too. The theorem we are using implicitly here states, in precise terms, that for every pair of infinite sequences and such that for every positive integer , , if the value of is infinite, so is the value of . That is, if , then . Even expressing it as precisely as this, it’s still pretty obvious that the theorem is true: for every positive integer , we clearly have , so it follows by one of the properties of limits that the same relation holds between the limits of these two sums, and the only thing greater than or equal to is itself.
The next manipulation we carried out, though, was perhaps more dubious: we said that since , , and so on, we could also write the series as
In precise terms, the theorem we’re using says that for every infinite sequence and every increasing infinite sequence of positive integers, if the value of (which we’ll call the condensed series) is infinite, so is the value of the original series. This theorem is actually a simple consequence of the fact that every sequence with a subsequence that diverges to also diverges to . With this fact in mind, all we need to do is note that for every positive integer , the sum of the first terms of the condensed series is simply , the sum of the first terms of the original series. So the sequence of partial sums of the condensed series is a subsequence of the sequence of partial sums of the original series (as long as is a strictly increasing sequence, which it is), and that completes the proof. Note that the theorem does not go the other way round: there are series of infinite value which can be condensed into series whose values are not infinite. For example, the harmonic series, which we know diverges to , contains as sub-series all the geometric series with common ratio where is a positive integer, and each such series converges to .
We then stated that the value of the series was obviously infinite; if you’re not happy with that “obviously”, just note that the sequence of partial sums here is , an arithmetic series beginning with 1 with the common difference , and all arithmetic series with a positive common difference diverge to .
The last thing was something we didn’t even explain in the original post, because it can’t really be explained without using the definition of infinite sums as limits. We said that if and are infinite sequences, is a finite real number , and for every positive integer , , then is a finite real number such that . The part is easy, because for every positive integer , we know that , so the same relation holds between their limits as approaches . What’s more tricky is proving that exists. The key is to note that the sequences of partial sums of and are both necessarily increasing, since all of their terms are non-negative. This means , being the limit of the sequence of partial sums of , must be greater than all of those partial sums. That means all the partial sums of are also less than . And it’s a basic fact about sequences that any increasing sequence whose terms never get greater than some real number must converge.