determine whether the series converges or diverges (without taylor expansion). ---**Textual Information:**
* **(f)** Mathematical Formula:
The sum from n equals 1 to infinity of (n squared multiplied by the tangent of (1 divided by n squared)).
**Mathematical Formulas/Chemical Equations:**
* (f) $$ \sum_{n=1}^{\infty} n^2 \tan\left(\frac{1}{n^2}\right). $$
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We need to determine whether the infinite series, sum from n equals 1 to infinity of n squared times tangent of 1 over n squared, converges or diverges. Let's analyze this series step by step without using Taylor expansion.