Estoy tratando de comprobar la convergencia o divergencia de la serie <span class="math-container">$\displaystyle\sum_{n=1}^{\infty}\dfrac1n\log\left(1+\dfrac1n\right)$</span>.
Mi intento: una finita <span class="math-container">$p$</span>,<span class="math-container">\begin{align}\displaystyle\sum{k=n}^{n+p}\dfrac1k\log\left(1+\dfrac1k\right)&\lt\dfrac1n\displaystyle\sum{k=n}^{n+p}\log\left(1+\dfrac1k\right)\&=\dfrac1n\log\large\Pi_{k=n}^{n+p}\left(\dfrac{k+1}{k}\right)\&=\dfrac1n\log\left(1+\dfrac{p+1}{n}\right)\&\lt\dfrac1n\log2,\text{ for large %#%#% and %#%#% is finite.}\&\lt\varepsilon\end {Alinee el}</span> por lo tanto la serie converge.