Assume I need to calculate the sum of the following power series: $$ S=\sum_{I=1}^\infty \frac{(-1)^n x^n y^n}{n} $$
The first way is to substitute: $$ t=xy \Rightarrow S= \sum_{I=1}^\infty \frac{(-1)^n t^n}{n} = -ln(1+t)=-ln(1+xy) $$
The second way is to substitute: $$ t=-xy \Rightarrow S=\int \sum_{I=1}^\infty t^{n-1} dt = \int \frac{1}{1-t} =ln(1+xy) $$
and the two solutions differ in a minus sign. I suspect something is wrong with the second way, but can' figure out what. WIll you please help ?
