:12ChapterInfiniteseriesSec.1 Definition and Properties ofInfinite Series
Chapter 12 Infinite series Sec.1 Definition and Properties of Infinite Series
S 12.1 Definition and Properties of Infinite SeriesI. Definition ofinfinite seriesOverview1181/1621684=82￥34The infinite sum was infinite.1-1+1-1+1-1+.:Theinfinite sumwas impossibleto pindown
§12.1 Definition and Properties of Infinite Series I. Definition of infinite series Overview 2 1 4 1 8 1 1/16 + + + + 16 1 8 1 4 1 2 1 = The infinite sum was infinite. 1−1+ 1−1+ 1−1+ The infinite sum was impossible to pin down. = 1 + + + + 4 1 3 1 2 1 1 1
S 12.1 Definition and Properties of Infinite SeriesL.Definition ofinfinite seriesGiven a sequence of numbers (un),The nth term80Zu,:=u, +u, +u +...+un +...n=1is called an infinite series of constant terms.Q: Whether theinfinite series has sum?
§12.1 Definition and Properties of Infinite Series I. Definition of infinite series = + + ++ + = n n un u1 u2 u3 u 1 is called an infinite series of constant terms. The nth term Given a sequence of numbers { }, un Q: Whether the infinite series has sum?
S 12.1 Definition and Properties of Infinite SeriesI.Definition of infinite seriesConsider the partial sum2S, = 1,1S1+2lim S. = 211n-→8S.=1十2221S=1+222321~122
§12.1 Definition and Properties of Infinite Series I. Definition of infinite series + + + ++ n + Consider the partial sum 2 1 2 1 2 1 2 1 1 2 3 1, S1 = , 2 1 S2 = 1+ , 2 1 2 1 1 3 2 S = + + 2 3 1 2 1 2 1 2 1 2 1 1 − = + + + + + n n S lim = 2 → n n S 1 2 1 2 − = − n
S 12.1 Definition and Properties of Infinite SeriesIl.Convergence of infinite seriesDefinition80We call the infinite seriesu, converges and hassum S if the sequence of partial sums fs, convergesto S. If s,diverges, then the series diverges.80IfZ1. converges, let r, = S-S, -Zu.lim r.n+k1n>8k=1n=1r, is called the remainder of the series.S ~ Sn, the error is r
§12.1 Definition and Properties of Infinite Series II. Convergence of infinite series Definition We call the infinite series converges and has sum S if the sequence of partial sums converges to S. If diverges, then the series diverges. n=1 un { } Sn { } Sn the error is . n , r S Sn If converges, let n=1 un , 1 = = − = + k n S Sn un k r = → n n lim r 0? rn is called the remainder of the series