Introduction to Infinite Series
Introduction to Infinite Series
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13. ^l trencral Tlworem. Ld "o + "i -r he any convergent series of2>ositive and negative terms. Then Lim u^^ = . More generally, Lim [//„ + ?/„ + ! + -f ;'„+, _i] = 0, ra = 00 irJiere ]) is any integer, either constant or varying irilh n.
The proof of this theorem How's directly out of the conception of a limit. Let and plot the points . Sj, . Sg, s^, Then what we mean when we say "x^ approaches a limit U" is that there is a point I about ■which the sja cluster, as n increases. This does not ne...cesi^arily re- quire that (as in the series hitherto eonsideied) s^ should always come steadily nearer to U, as n increases. Thus s.^ may lie further away from [' than . So does. But it does mean that ultimately the sjs will S3 U-c^ s^ U s^U+c^ s, \ \ 1 Mill 1 1 \ — Fk;. 7.
cease to deviate from U liy more than any arbitrarily assigned quan- tity, S, however small. In other words, let 8 be taken at pleasure (rr 1/1, 000, 000, say) and lay off an interval extending to a dis- tance 8 from U in each direction, ((/ — 8, f'-j-8); then for the larger values of ?«, more precisely, for all values of u greatei- than a certain fixed number m, s^ will lie Avithin this interval.
What to read after Introduction to Infinite Series?
You can find similar books in the "Read Also" column, or choose other free books by William F William Fogg Osgood to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
13. ^l trencral Tlworem. Ld "o + "i -r he any convergent series of2>ositive and negative terms. Then Lim u^^ = . More generally, Lim [//„ + ?/„ + ! + -f ;'„+, _i] = 0, ra = 00 irJiere ]) is any integer, either constant or varying irilh n.
The proof of this theorem How's directly out of the conception of a limit. Let and plot the points . Sj, . Sg, s^, Then what we mean when we say "x^ approaches a limit U" is that there is a point I about ■which the sja cluster, as n increases. This does not ne...cesi^arily re- quire that (as in the series hitherto eonsideied) s^ should always come steadily nearer to U, as n increases. Thus s.^ may lie further away from [' than . So does. But it does mean that ultimately the sjs will S3 U-c^ s^ U s^U+c^ s, \ \ 1 Mill 1 1 \ — Fk;. 7.
cease to deviate from U liy more than any arbitrarily assigned quan- tity, S, however small. In other words, let 8 be taken at pleasure (rr 1/1, 000, 000, say) and lay off an interval extending to a dis- tance 8 from U in each direction, ((/ — 8, f'-j-8); then for the larger values of ?«, more precisely, for all values of u greatei- than a certain fixed number m, s^ will lie Avithin this interval.
What to read after Introduction to Infinite Series?
You can find similar books in the "Read Also" column, or choose other free books by William F William Fogg Osgood to read onlineMoreLess
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