Question

幂级数求和问题，求指教：∑（n从1到正无穷）(n^2+1)/n * x^2n 不胜感激！！！

Answer 1

∑（n从1到正无穷）[(n²+1)/n ] x^(2n)
=∑（n从1到正无穷）nx^(2n)+∑（n从1到正无穷）(1/n)x^(2n)
=x/2∑（n从1到正无穷）2nx^(2n-1)+2∑（n从1到正无穷）[x^(2n)]/2n
=x/2∑（n从1到正无穷）[x^(2n)]′+2∑（n从1到正无穷）∫x^(2n-1)dx(积分区间为0到x)
=x/2[∑（n从1到正无穷）x^(2n)]′+2∫[∑（n从1到正无穷）x^(2n-1)]dx
=x/2[x²/(1-x²)]′+2∫[x/(1-x²)]dx
=x²/(1-x²)²-ln|1-x²|