Taylor's theorem
a) If $x∈{0,1}$ and $n∈ℕ$, show that $$ \begin{vmatrix} ln(1+x)-(x-\frac{x^{2}}{2}+\frac{x^{3}}{3}+...+(-1)^{n-1}\frac{x^{n}}{n}) \\ \end{vmatrix}<\frac{x^{n+1}}{n+1} $$
b)Using the result of part "a" to Show $\sum_{n=1}^∞ \frac{(-1)^{n-1}}{n}=\ln(2)$ Note: If $(-1)^{0}=1$
I already proved part a, but I do not know how to do part b although I know it is simpler than a. If your could help me, I'll thank you
One can check the convergence of the series by the alternating series test. Abel's theorem then tells us that the value is continuous from the left, meaning the value is $\ln(1 + 1)$.