微积分作业与答案作业29.pdf

1、按定义判断下列级数的敛散性，若收敛，求其和

∞1

1）∑

n1(5n−4)(5n+1)

1111111111

解：s=++L+=−+−+L+−

n

1661154515166115451

××n−n+n−n+

()()

11

1=−

55n+1

111

limslim1=−

n

n→∞n→∞55n+15

∞1∞11

级数∑收敛，∑

n1(5n−4)(5n+1)n1(5n−4)(5n+1)5

∞1

2）∑

n++n

n11

1n+1−n∞

解：因为n+1+n1，原级数化为∑(n+1−n)

n1

s2=−1+3−2+L+n+1−nn=+1−1

n

limsnlim(n=+1−1)=∞

n→∞n→∞

∞1

级数∑发散。

n1n

n1++

∞

3）∑(n+2−2n+1+n)

n1

解：n+−n++nn=+−n+−n+−n

221(21)(1)

s3=−2−2−1+4−3−3−2+L+n+2−n+1−n+1−n

n()()()()()()

1

n=+2−n+1−2−11=−2+

()n+2+n+1