How Can I Express This Alternating Pattern Series as a Single Sum?

[Nick89]

Homework Statement

I need a way to write the following series in a sum:
$$\frac{1}{2} + \frac{1}{4} + \frac{1}{32} + \frac{1}{64} + \frac{1}{1024} + \frac{1}{2048} + ...$$

If you look closely you can see a 2^n pattern in there:
$$\frac{1}{2} + \frac{1}{4} + \frac{0}{8} + \frac{0}{16} + \frac{1}{32} + \frac{1}{64} + \frac{0}{128} + \frac{0}{256} + \frac{1}{512} + \frac{1}{1024} + ...$$

The denominator obviously follows $$2^n$$, but the numerator goes 110011001100...
I can't seem to find a function that will allow me to put this in a single sum:
$$\sum_{n=1}^{\infty} ...$$

(The sum should converge to 0.8, right?)

The Attempt at a Solution

I tried using the mod operator to determine if n was even or odd, something like this:
$$\sum_{n=1}^{\infty} \frac{n \mod 2}{2^n} + \frac{n \mod 2}{2^{n+1}}$$
This doesn't work, because the terms that are discarded when n is even are not discarded the next 'run' when (n+1) is even... Dunno how to explain this properly, but if you calculate it manually it does this:
$$\frac{1}{2} + \frac{1}{4} + \frac{0}{4} + \frac{0}{8} + \frac{1}{8} + \frac{1}{16} + ...$$
So if you discard the 0/ ... terms you are just left with the usual 2^(-n) sum...

I have a feeling I'm close, but I can't figure it out :S

 

[epenguin]

If the missing bits were not missing it would add up to 1.

Now you can put each missing bit into a relation with a bit that is there.

They are a certain fraction of terms that are there. So their sum is that proportion of 1 IYSWIM.

Do a diagram and this will look plausible.

 

[Nick89]

I can't seem to understand what you mean exactly...

By missing bits you mean the 0/x parts?
If so, then half of the terms as missing, so to your logic that would mean the sum would converge to 0.5...? (Whatr does IYSWIM mean?)

But I think it should converge to 0.8 (I said 8 before but that was a mistake, I edited it now).

 

[Dick]

If it's the pattern you suggest, (1/2+1/4)*(1/16)=(1/32+1/64). (1/32+1/64)*(1/16)=(1/512+1/1024). It's a geometric series with a common ratio of 1/16. But this doesn't quite fit with the series you quote in the first line (the next term after 1/64 is 1/1024, not 1/512). Is that a typo?

 

[epenguin]

I can't seem to understand what you mean exactly...

By missing bits you mean the 0/x parts?
If so, then half of the terms as missing, so to your logic that would mean the sum would converge to 0.5...? (Whatr does IYSWIM mean?)

But I think it should converge to 0.8 (I said 8 before but that was a mistake, I edited it now).

IYSWIM means 'if you see what I mean'.
Just compare each pair of 'absent' terms with the 'present' pair preceding it.

 

[Nick89]

If it's the pattern you suggest, (1/2+1/4)*(1/16)=(1/32+1/64). (1/32+1/64)*(1/16)=(1/512+1/1024). It's a geometric series with a common ratio of 1/16. But this doesn't quite fit with the series you quote in the first line (the next term after 1/64 is 1/1024, not 1/512). Is that a typo?

It was indeed a typo. The correct form is right underneath, where 1/512 is present.

Alright, that makes a bit more sense. I still fail to see how I can now create a single sum for this series though... If I have a bit more time (maybe later tonight) I'll try to figure it out :)

Thanks so far.