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

Series
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asked 2021-03-05

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

Expert Answers (1)

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.

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