Where Does This Series Go?

One of my favorite series is the harmonic series.

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + …

As visible, the terms in the sequence get smaller and smaller.  One of the most intuitive questions to ask thus is whether the sum of the sequence (the series) converges or diverges (converge means to approach a certain value as infinite terms are added, and diverge means for the series to be able to reach any infinite value if enough terms are added).  Regardless of your experience with advanced mathematics, and calculus, this question is an intriguing one, and basic intuition might tell us that since each term is smaller that the previous, this series may converge.

wave-forms-of-non-linear-harmonic-series.jpg

Well the truth is that the series goes on to infinity, and therefore never converges.  The proof for this was quite captivating to me, and I wanted to share it, as I think it can appeal to people irrespective of their mathematical background.

If we want to show that something adds up to infinity, substituting values that are less than the real ones is a way to show divergence, as if that new series which we know is less than the real one adds up infinitely, we can be confident the real one does too.

So let’s break it down:

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/4 + 1/15 + 1/16 +…

>

1/1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + …

Do you see what happened here? Each term was 1/n was replaced by 1/2k where 2k is the smallest power of 2 greater than or equal to n. This means that each term was replaced by a term less than or equal to it, and so let’s try and see what happens to this new series.

1/1 + (1/2) + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + (1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16) + …

As seen, we can group sets of terms into terms that add up to half. Since there are infinitely many terms, we can add infinitely many halves, and if you infinitely, add up 1/2, you will never reach a specific value, and will add up infinitely. Pretty cool, right?

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