∑(∞,n=1)2nx^(2n-1)/(2n-1)收斂域及和函式

1.收斂域

顯然收斂區間為(-1,1)

2nx^(2n-1)/(2n-1)=(2n-1+1)x^(2n-1)/(2n-1)=x^(2n-1)+x^(2n-1)/(2n-1)

∑(∞,n=1)x^(2n-1)在x=±1時發散,所以

收斂域為(-1,1)

2.和函式

2nx^(2n-1)/(2n-1)=(2n-1+1)x^(2n-1)/(2n-1)=x^(2n-1)+x^(2n-1)/(2n-1)

∑(∞,n=1)2nx^(2n-1)/(2n-1)=

∑(∞,n=1)x^(2n-1)+∑(∞,n=1)x^(2n-1)/(2n-1)=s1+s2

s1=x/(1-x^2)

∑(∞,n=1)x^(2n-1)/(2n-1)=s2

s2"=[∑(∞,n=1)x^(2n-1)/(2n-1)]"

=∑(∞,n=1)[x^(2n-1)/(2n-1)]"

=∑(∞,n=1)x^(2n-2)

=1/(1-x^2)

s1=∫(0到x)s1"dx=∫(0到x)1/(1-x^2)dx=1/2ln|(1+x)/(1-x)|

所以冪級數∑(∞,n=1)2nx^(2n-1)/(2n-1)收斂域及和函式為

s(x)=x/(1-x^2)+1/2ln|(1+x)/(1-x)|.