The Algebra of Coplanar Vectors And Trigonometry

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is convergent, still more is the general series convergent. If the chain stretched out to its full length, (that is, so that the links are all in the same line,) has a finite length, any coiling or folding must bring its ends nearer together.
Hence, if the scalar series, which we may call the r-series, is convergent for all values of r, so is also the general series, which we may call the (r, u) series.
More commonly however the r-series is convergent only when r is less (or not greater) th
...an some fixed quantity k ; then the (r,u) series is certainly convergent when r is less (or not greater) than k, but it may be that when r = k, the (r, u) series is convergent, though the r-series is not so. We shall also show (what is not in itself obvious) that it is divergent, when r is greater than k.
It would seem that this last assertion is not necessarily true, when the coefficients are complex. Though the regular coiling (due to the versors, ■*'*, i 2u , ifec.) may not bring the ends of the infinite chain to a finite distance from one another, it is con- ceivable that some farther folding (due to the versors i% i™-..-) might do so.


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