Sequences and series (Library of mathematics) by John A. Green

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The logarithmic series. •. IUn+llunl=l-xnln+rl=lxl{nln+r)~lxl, as 3 4 Therefore the series converges if Ixlr. Thus the radius of convergence is 1. {We shall see (p. ) Example 5. The binomial series. By putting y=r in the binomial theorem (p. 25) we obtain the finite series {r+x)1I n~oo. = ~ (=)xn. Now the 'binomial coefficient' (:) can be de- fined, even when a is not a positive integer, by the usual for) (a) =r. H ow· mu Iae ( a) a{a-r) ... 2 . • . n 0 ever, in this case the series I (:)xn will be infinite, because n-O the coefficients (=) are all non-zero, unless a is a positive integer or zero.

Only to find his opponent leading, by the small-but positive -distance 11m"? The answer is that the whole infinite sequence of stages T oT 1 , T IT 2, T 2T a, •.. •• is finite, even though it is composed of an infinity of parts. At the end of this finite length a point U is reached where the runners are level. After that, Achilles takes the lead. The distance ToU is given by the infinite sum 1+{l/m)+{l/m)2+ ••• , and we shall see in § 5 how to evaluate this. 3. CONVERGENT AND DIVERGENT SERIES L: <0 If we want to find the 'sum' of the infinite series tI.

Converges also. (it) If p" >q"for all n, and ij"Eq" diverges, then "Ep,. diverges also. ) and (t,,) of partial sums ... +p", ttl =ql +qa + ... +q" are both increasing. ---+ a limit T. All the ttl are < T. But p,, q", for all n, hence 5,,> ttl for all n, and thus srI must tend to + 00 as well.