《高等数学》课程电子教案（PPT课件）Chapter 12 Infinite series Sec 1 Definition and Properties of Infinite Series

: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

S 12.1 Definition and Properties of Infinite SeriesIl.Convergence ofinfinite seriesE.g.1 Show that a geometric series80Z= a +aq+aq +...+aq"-1 +..(a+ 0)aq'n=1converges if q n->00when q>1,lims,=o0limgn=0divergesn-00n>o

§12.1 Definition and Properties of Infinite Series II. Convergence of infinite series convergesif 1,but diverges if 1. ( 0) Show that a geometric series 2 1 1 1   = + + + + +  −  = −  q q aq a aq aq aq a n n n   E.g.1 Solution: 2 −1 = + + + + n sn a aq aq  aq q a aqn − − = 1 q aq q a n − − − = 1 1 when 1, q  lim = 0 → n n  q q a sn n −  = → 1 lim when 1, q  =  → n n  limq  =  → n n lim s converges diverges

S 12.1 Definition and Properties of Infinite SeriesIl.Convergenceofinfinite seriesE.g.1 Show that a geometric series80Z= a+ aq+aq +...+aq"-- +...(a+ 0)aq'n=1converges if q <1, but diverges if q| ≥1.Solution:when q=l, Sn = na →0divergeswhen q=-l, the seriesis, a-a+a-a+.lim s, doesn't exist.divergesn→808MSo,aq"-1 convergesif |al <1, but diverges if || ≥1.n=l

§12.1 Definition and Properties of Infinite Series II. Convergence of infinite series E.g.1 Solution: when 1, q = when 1, q = − sn = na →  convergesif 1,but diverges if 1. ( 0) Show that a geometric series 2 1 1 1   = + + + + +  −  = −  q q aq a aq aq aq a n n n   the seriesis, a − a + a − a + lim doesn't exist. n n s →  diverges diverges So, convergesif 1,but diverges if 1. 1 1     = − aq q q n n

S 12.1 Definition and Properties of Infinite SeriesIl.Convergenceofinfinite series2元,Z5", Zk, Z11E.g.2n=1n=1n=1[=[5|>1Solution:28XZ5nconverges and diverges.2nn=1n80.Zlim S. = 00S, = nk,k diverges.n-8n=1T, =12 +22 +...+nlim T.=8n->008N,n? diverges.n=1

§12.1 Definition and Properties of Infinite Series II. Convergence of infinite series E.g.2 , 5 , 2 1 1 1    =  = n n n n , 1   n= k 1 2   n= n 1, 5 1 2 1 q1 =  q2 =  converges and 5 diverges. 2 1 1 1    =  =  n n n n Solution: S nk, n = =  → n n lim S diverges. 1   =  n k 2 2 2 Tn = 1 + 2 ++ n =  → n n lim T diverges. 1 2   =  n n

S 12.1 Definition and Properties of Infinite SeriesII. Convergence of infinite series81ZE.g.3n(n + 1)1111Solution:S1.22.3n·(n +1)111232n+1n1nn+1n+1文1lim S., = 1Zn-→n(n+ 1)=i

§12.1 Definition and Properties of Infinite Series II. Convergence of infinite series ( 1) 1 1   n= n n + E.g.3 Solution: ( 1) 1 2 3 1 1 2 1  + + +  +  = n n Sn  1 1 1 3 1 2 1 2 1 1 + = − + − + + − n n  1 1 1 1 + = + = − n n n lim = 1 → n n S 1 ( 1) 1 1 = +   n= n n

S 12.1 Definition and Properties of Infinite SeriesIl.ConvergenceofinfiniteseriesE.g.4 A Bouncing BallYou drop a ballfrom a meters above a flat surface.Eachtime the ball hits the surface afterfalling a distance h, itrebounds a distance rh,whereris positive bur lessthan 1.Find thetotal distancethe ball travels up and down.aar2ar4arar4

§12.1 Definition and Properties of Infinite Series II. Convergence of infinite series E.g.4 A Bouncing Ball You drop a ball from a meters above a flat surface. Each time the ball hits the surface after falling a distance h, it rebounds a distance rh, where r is positive bur less than 1. Find the total distance the ball travels up and down. ar2 ar3 ar 4 ara