This image appears to contain mathematical expressions related to series or sequences, possibly in the context of convergence tests like the Cauchy convergence criterion. To analyze it using Cauchy’s convergence criterion, we must look at how the terms behave as the series progresses.

Cauchy's criterion states that a series $\sum_{n=1}^{\infty} a_n$ converges if, for every $\epsilon > 0$, there exists an integer $N$ such that for all $m, n \geq N$, the partial sum of the terms satisfies:

$\left| \sum_{k=n}^{m} a_k \right| < \epsilon$

If you can provide the sequence or series in clearer detail or ask a specific question about it, I can help you better apply the Cauchy criterion. Would you like me to provide further analysis or check specific steps?