Prove Divergent Series of Positive Sequence | MathHelpForum.com

[chisigma]

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A constructive divergent series

Let $\displaystyle \{x_{n}\}$ be a positive sequence satisfying $\lim_{\ n \rightarrow \infty} x_{n}= \infty$ Prove that the series $\displaystyle \sum_{n=1}^{\infty} (1-\frac{x_{n}}{x_{n+1}})$ is divergent.

Till now no answers have been given. The solution I propose involves an 'infinite product'. It is well known the theorem that establish that the 'infinite product' $\displaystyle \prod_{n=1}^{\infty} (1-a_{n})$ diverges if the 'infinite sum' $\displaystyle \sum_{n=1}^{\infty} a_{n}$ diverges. Less known but equally valid is the converse theorem that establishes that the 'infinite sum' $\displaystyle \sum_{n=1}^{\infty} a_{n}$ where $\displaystyle 0<a_{n}<1$ diverges if the 'infinite product' $\displaystyle \prod_{n=1}^{\infty} (1-a_{n})$ diverges to 0 as You can see here...

http://en.wikipedia.org/wiki/Convergence_tests

Applying the converse theorem and setting $\displaystyle a_{n}= 1- \frac{x_{n}}{x_{n+1}}$ we obtain that...

$\displaystyle \prod_{n=1}^{\infty} (1-a_{n}) = \prod_{n=1}^{\infty} \frac{x_{n}}{x_{n+1}}= \frac{x_{1}}{x_{2}}\ \frac{x_{2}}{x_{3}}\ ...$ (1)

Now the product (1) is 'telescopic' and is...

$\displaystyle p_{k}= \prod_{n=1}^{k} \frac{x_{n}}{x_{n+1}}= \frac{x_{1}}{x_{k+1}}$ (2)

... and, because is $\displaystyle \lim_{\ k \rightarrow \infty} x_{k+1}=\infty$ is also $\displaystyle \lim_{\ k \rightarrow \infty} p_{k}=0$, the product (1) diverges and for the converse theorem the series $\displaystyle \sum_{n=1}^{\infty} (1-\frac{x_{n}}{x_{n+1}})$ diverges...

Kind regards

$\chi$ $\sigma$

 

[jvicens]

Dear $\chi$ $\sigma$,

Thank you for sharing your solution to this problem. It is always interesting to see different approaches to solving mathematical problems. I would like to add some additional thoughts to your solution.

Firstly, your use of the converse theorem is a valid approach to proving the divergence of this series. However, it may not be immediately obvious to everyone how the converse theorem applies in this situation. It may be helpful to provide a brief explanation or proof of the converse theorem for those who may not be familiar with it.

Secondly, while your solution is correct, it may be helpful to provide some additional justification for your statement that the product (1) is "telescopic." This term may not be familiar to all readers and it would be beneficial to explain how the product simplifies to $\frac{x_1}{x_{k+1}}$.

Overall, your solution is well-written and provides a clear and concise explanation for the convergence of the series. Thank you for contributing to the discussion on this problem.