Proceedings of the Royal Society of London
Proceedings of the Royal Society of London
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In our eReader you can find the full English version of the book. Read Proceedings of the Royal Society of London Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book Proceedings of the Royal Society of London
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+ u^^-^- ... be any series, then X call ^-^^ when n increases without limit the ultimate ratio of the series, and denote it by p. Then if or is less than -, the series will be convergent.
Now consider the function - - which we have employed in these investigations.
Since r + — tTT ^^ ^^ series comprising can contain no even powers, and therefore we may assume — — «tt^ + Wja? + tt,a?* + tt,a?»+ .. .
Expanding e'+l in terms of x^ and multiplying the series together, we obtain ^+^ + t:2 +1717374+ ...••••"^' orl + l+^^-l- + !^\^^.— i— u^j^i 1.2 t42»-i tt3»+i 1.2.3.4 Let n increase without limit, and then 14a»_l U9n-S Digitized by VjOOQ IC lerr.] Pre$eHt«. 86S whence 1 + cos -i « 0, which gives 111 .p._-j,or-gp,or-^....; that is to say, the series will be conyergent if —^ — , —-r ... are all less than unity — ^that is, if « is less than w ; which accords with the result given by Cauchy.
Let us next consider the series «^+' « u^ + t»i^ + tt^ + . . . , we easily obtain by differentiation the following equation for deter- mining tf|„ (n+1) u„+i « u, + 2u««i, which may be written As n increases without limit, — ^ » ap, and the equation becomes p'a 0, or p B 0, whence the series is conyergent however great x may become.
What to read after Proceedings of the Royal Society of London?
You can find similar books in the "Read Also" column, or choose other free books by Royal Society (Great Britain) to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
+ u^^-^- ... be any series, then X call ^-^^ when n increases without limit the ultimate ratio of the series, and denote it by p. Then if or is less than -, the series will be convergent.
Now consider the function - - which we have employed in these investigations.
Since r + — tTT ^^ ^^ series comprising can contain no even powers, and therefore we may assume — — «tt^ + Wja? + tt,a?* + tt,a?»+ .. .
Expanding e'+l in terms of x^ and multiplying the series together, we obtain ^+^ + t:2 +1717374+ ...••••"^' orl + l+^^-l- + !^\^^.— i— u^j^i 1.2 t42»-i tt3»+i 1.2.3.4 Let n increase without limit, and then 14a»_l U9n-S Digitized by VjOOQ IC lerr.] Pre$eHt«. 86S whence 1 + cos -i « 0, which gives 111 .p._-j,or-gp,or-^....; that is to say, the series will be conyergent if —^ — , —-r ... are all less than unity — ^that is, if « is less than w ; which accords with the result given by Cauchy.
Let us next consider the series «^+' « u^ + t»i^ + tt^ + . . . , we easily obtain by differentiation the following equation for deter- mining tf|„ (n+1) u„+i « u, + 2u««i, which may be written As n increases without limit, — ^ » ap, and the equation becomes p'a 0, or p B 0, whence the series is conyergent however great x may become.
What to read after Proceedings of the Royal Society of London?
You can find similar books in the "Read Also" column, or choose other free books by Royal Society (Great Britain) to read onlineMoreLess
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