Summation Analysis: Evaluate $ \sum \frac{n^2}{3n^2 + 1} $

Math Problem Statement

Evaluate the summation $ \sum \frac{n^2}{3n^2 + 1} $.

The given expression is a summation of the form:

$\sum \frac{n^2}{3n^2 + 1}$

To analyze this, let's break it down:

The General Term: The term being summed is $\frac{n^2}{3n^2 + 1}$ .
Convergence/Divergence: Depending on the range of summation, we may need to analyze whether the series converges or diverges. For large $n$ , the term simplifies approximately to $\frac{n^2}{3n^2} = \frac{1}{3}$ , which indicates the terms do not shrink fast enough for the series to converge (if the summation is from $n = 1$ to $\infty$ ).

Could you confirm the summation limits (e.g., from 1 to infinity, or any other bounds)?

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Series and Summation
Convergence and Divergence
Asymptotic Behavior

General term: $ \frac{n^2}{3n^2 + 1} $
Approximation for large $ n $: $ \frac{n^2}{3n^2} \approx \frac{1}{3} $

Divergence Test for Series

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