Show that
![\[ T_n ( \log (1+x)) = \sum_{k=1}^n \frac{(-1)^{k+1} x^k}{k}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-33fe7dc1f6fa58eedb11127d50ae5184_l3.png "Rendered by QuickLaTeX.com")
Since
![\[ \log (1+x) = \int \frac{1}{1+x} \, dx \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-8fcd760812abc6c8e9c8f54753689c2a_l3.png "Rendered by QuickLaTeX.com")
we can apply Theorem 7.2 (c) (page 276 of Apostol) and our result from this exercise to compute,
![\begin{align*} T_n (\log (1+x)) &= \int T_{n-1} \left( \frac{1}{1+x} \right) \, dx \\[9pt] &= \int \sum_{k=0}^{n-1} (-1)^k x^k \, dx \\[9pt] &= \sum_{k=0}^{n-1} (-1)^k \frac{x^{k+1}}{k+1} \\[9pt] &= \sum_{k=1}^n \frac{(-1)^{k+1} x^k}{k}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-e3387a0b076cbef7cb67f57b0edb3040_l3.png "Rendered by QuickLaTeX.com")
Show that
Since
we can apply Theorem 7.2 (c) (page 276 of Apostol) and our result from this exercise to compute,