Question

# Q1. Does a series sum_infty^{n=1}bn converge if bnrightarrow 0? Justify your answer by at least two examples?

Series

Q1. Does a series $$\displaystyle{\sum_{\infty}^{{{n}={1}}}}{b}{n}$$ converge if $$bn \rightarrow 0$$? Justify your answer by at least two examples?

2021-03-06

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.