Limit of $a_{n+1} = a_{n}(1+1/n^2)$

As $\ln x$ is increasing, $b_n = \ln a_n$ is an increasing sequence.

Also $$b_{n+1}= \sum_{k=1}^n \ln(1+1/k^2) \le \sum_{k=1}^n 1/k^2$$

according to the provided hint.

As the series on the RHS is convergent, the sequence $\{b_n\}$ is bounded. Being also increasing, it is convergent. Therefore so is $\{a_n\}$.

Tags:

Limits

Sequences And Series

Real Analysis

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