Questions: We have seen that the harmonic series is a divergent series whose terms approach 0. Show that [ sumn=1^infty ln left(1+frac1nright) ] is another series with this property.

Solution Steps

We consider the series

\[ \sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right). \]

For large \( n \), we can use the approximation

\[ \ln \left(1+\frac{1}{n}\right) \approx \frac{1}{n}. \]

The series

\[ \sum_{n=1}^{\infty} \frac{1}{n} \]

is known to be divergent. Since

\[ \ln \left(1+\frac{1}{n}\right) \approx \frac{1}{n} \]

for large \( n \), we can conclude that

\[ \sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \]

behaves similarly to the harmonic series and is therefore also divergent.

Final Answer