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Thread: can someone please explain to me the telescoping series(1+2+4+8+16...) and how it equals to -1

  1. #1 can someone please explain to me the telescoping series(1+2+4+8+16...) and how it equals to -1 
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    im having alot of trouble with this concept please help


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    Forum Senior TheObserver's Avatar
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    That sum does not converge and so cannot telescope. I don't know where you got this -1 business from.


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    Forum Freshman Saturn's Avatar
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    I'm also unsure of how you got -1. Since you say you're having trouble with the concept of a telescoping series, this video might help. It shows an example of a pretty simple one that converges (there's a link in the description to a diverging example as well). He has a ton of really useful math videos on his website too, if there are other topics you're struggling with. Hope it helps.
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    Forum Professor river_rat's Avatar
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    Hi incorrect

    There is a whole section of mathematics devoted to making sense of divergent series and it has many important applications. But remember these are all about extending definitions and so we are already out of the realm of normal infinite series, so keep your wits about you. Also, it using analytic continuation so if you have a shaky complex analysis foundation this may seem like cheating.

    The result you have can be derived as follows. Consider the complex function defined as a power series. We know that this power series has radius of convergence < 1 and so defines a perfectly reasonable analytic function in that disc. Now this is also a geometric series and so we have the following identity when . So lets see what we have, a honest to goodness analytic function which happens to be equal to another honest to goodness meromorphic function on the disc . But is actual defined on the entire complex plane except for a pole at and thus we can use it to extend the definition of our original function outside of its radius of convergence. This is the analytic continuation part.

    But note what we have done here, we have in a sense defined formally for all z now. Substitute for and we get the formal identity you asked about We have not actually done the sum in the sense that the partial sums converge to -1, as each partial sum is positive so is always far away from -1. We have instead extended our definition of what an infinite sum could mean using the concept of analytic continuation and this distinction is important.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Forum Professor jrmonroe's Avatar
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    The series of 1+2+4+8+... equals 1 if the binary word resulting from the series (eg, the 16-bit word 1111111111111111) is interpreted using the two's complement convention.
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    Forum Professor river_rat's Avatar
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    hi jrmonroe

    The series 1 + 2 + 4 + 8 + ... doesn't have a representation in binary as you would have constant overflow and loop round (assuming unsigned types)?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  8. #7  
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    Quote Originally Posted by river_rat View Post
    hi jrmonroe

    The series 1 + 2 + 4 + 8 + ... doesn't have a representation in binary as you would have constant overflow and loop round (assuming unsigned types)?
    There would be no overflow since it's a sum of powers of 2 and it's not unsigned, by definition, if it has a two's complement representation. Then again, it is a ridiculous idea.
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