f′(x)=ln(1+x)+1
=[∑(n从1到∞)(-1)^(n-1)x^n/n]+1
f(x)=∫(0到x)f′(x)dx+f(0)
=∫(0到x){[∑(n从1到∞)(-1)^(n-1)x^n/n]+1} dx
=∫(0到x)∑(n从1到∞)(-1)^(n-1)x^n/ndx+x
=x+∑(n从1到∞)(-1)^(n-1)∫(0到x)x^n/ndx
=x+∑(n从1到∞)[(-1)^(n-1)/n(n+1)]x^(n+1)
f(x)=(1+x)ln(1+x)展开成x的幂级数
f′(x)=ln(1+x)+1
=[∑(n从1到∞)(-1)^(n-1)x^n/n]+1
f(x)=∫(0到x)f′(x)dx+f(0)
=∫(0到x){[∑(n从1到∞)(-1)^(n-1)x^n/n]+1} dx
=∫(0到x)∑(n从1到∞)(-1)^(n-1)x^n/ndx+x
=x+∑(n从1到∞)(-1)^(n-1)∫(0到x)x^n/ndx
=x+∑(n从1到∞)[(-1)^(n-1)/n(n+1)]x^(n+1)