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What is this sequence?


ajinx999Oct 2, 2007 3:17am
This is like a mathematical paradox. What's the incongruity in it?

The sequence is
1, -1/2, 1/3, -1/4, 1/5, -1/6 ...
The question is to find the value of its series.

This is surely a convergent sequence. First, I'll go the "straight" general way
(1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + ... = 1/2 + 1/12 + 1/30 + 1/56 ...
Therefore, after perusing the RHS (right-hand-side), the value of the series on LHS comes out to be greater than 1/2.

Now, here is the paradox-like
Let the value of the series be S
S = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...
= (1 + 1/3 + 1/5 + 1/7 + ...) - (1/2 + 1/4 + 1/6 + 1/8 + ...)
= (1 + 1/3 + 1/5 + 1/7 + ...) - (1/2 + 1/4 + 1/6 + 1/8 + ...) + (1/2 + 1/4 + 1/6 + 1/8 + ...) - (1/2 + 1/4 + 1/6 + 1/8 + ...)
I added and subtracted (1/2 + 1/4 + 1/6 + 1/8 + ...)
S = (1 + 1/2 + 1/3 + 1/4 + ...) - 2(1/2 + 1/4 + 1/6 + 1/8 + ...)
= (1 + 1/2 + 1/3 + 1/4 + ...) - (1 + 1/2 + 1/3 + 1/4 + ...)
= 0
Where's the anomaly?

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MorosophOct 2, 2007 9:49am
The answer is that the terms in brackets are divergent series. You could make the above series sum to any real (or even ±∞) by re-ordering the elements; the series is valid only as long as there is a finite upper bound upon the distance that an element is moved.

What you've done here involves a kind of re-ordering in that evaluating 2(1/2 + 1/4 + 1/6 + 1/8 + ...) has moved the elements of the series higher up the sequence in their evaluation.

More mundanely, the calculation beyond the first step isn't valid, since it involves differences between infinities, which does not give a well-defined result.

ajinx999Oct 2, 2007 8:16pm
@Morosoph:
Thanks! You cleared my doubt, almost!
The sequence is divergent because the subsequences don't have a common finite limit. Is this what you meant? I, probably, misinterpreted the meaning of convergence by considering that the sequence has a finite value.
The series of the given sequence is actually ln2.

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MorosophOct 3, 2007 6:59am
No, I mean that the bracketed sub-series are divergent. The series as a whole is not. However, if you insert terms from the sub-series out-of order, you can sum to any number you like! To do this, choose a real number. To form your new series, pick the next element from the ascending series when below your chosen number, and pick from the descending series when above it. Repeat ad infinitum.

A quick point of terminology: when you sum the terms, it is called a series. A sequence is one element followed by another, typically comma-separated.

I thought that I recognised the ln2 series from somewhere. Neat trick, BTW!

ajinx999Oct 5, 2007 5:59am
@Morosoph:
I know that terminology. If I didn't knew that then how would you expect me to know terms like subsequence.
I understood what you meant about the divergence of the bracketed series.
The given alternating harmonic series is convergent but the bracketed series (1+ 1/2 + 1/3 + 1/4 + ...) is divergent. Am I right?
Thanks!

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MorosophOct 5, 2007 6:28am
Bingo!

Sorry, I wasn't meaning to "lord it over" or anything, but it is (I believe) quite a common error to call series sequences. It's worth remembering that we have an audience, so although it's a minor point, it's worth correcting :o)

ajinx999Oct 5, 2007 8:00am
@Morosoph:
OK, I'm glad you look over such small facts. I also follow that way. Being purist is no harm! It depicts perfection.
You said "it's worth correcting" Did I go wrong somewhere?

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MorosophOct 5, 2007 8:14am
No, I just read you too fast. I tripped up when I read:

    This is surely a convergent sequence. First, I'll go the "straight" general way
    (1 - 1/2) + ...

My mistake.

ajinx999Oct 5, 2007 8:28am
@Morosoph:
That was a sarcastic remark, I suppose. But I got the error. The way I stated it created misunderstanding. I should have attached that statement to my first para.


What is this sequence?

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