Here are a couple of easier examples so that you get a better handle on this.

A geometric series

Consider for example \(\displaystyle{b}{n}={\frac{{{1}}}{{{2}^{{{n}}}}}}\). Clearly \(bn\rightarrow0\) as \(n\rightarrow \infty\). Then the series

\(\displaystyle{\sum_{\infty}^{{{n}={1}}}}{b}{n}\)

This result can be shown geometrically (hence the name of this type of series) or by rewriting the series as a telescoping sum. So clearly this series satisfies the given condition and converges.

The harmonic series

Now consider the very simple looking example \(\displaystyle{b}{n}={\frac{{{1}}}{{{n}}}}\). Again, it should be easier to see that \(bn\rightarrow 0\) as \(n\rightarrow \infty\). However, this time the series diverges to infinity. This can be shown using either the comparison test or the integral test.

Conclusion

Thus the requirement that \(bn\rightarrow 0\) is not a strong enough requirement to let us know whether a series converges or not. There are examples where it does and examples where it doesn't.