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Convergence in terms of infinite series refers to the behavior where the sum of an infinite sequence of numbers approaches a specific value as more and more terms are added. It's like walking closer to a fixed point. In mathematical terms, for a series to converge, the partial sums of the series must get closer and closer to a definite value. If it does, the series is called convergent.

In our exercise, the series involves terms like \( \frac{1}{n^2} \). As \( n \) becomes very large, each term \( \frac{1}{n^2} \) becomes very small, essentially shrinking towards zero. This shrinking effect is crucial, as it suggests that adding such small values continually results in a sum that settles near a certain number. This is the essence of convergence in our context. For the given series, this specific target value or sum lies between certain bounds, such as \( \frac{17}{12} \) and 2, indicating its convergent nature.

Comparison of fractions is foundational in understanding inequalities within infinite series. When comparing fractions like \( \frac{1}{3^2} \) and \( \frac{1}{2^2} \), think of the denominators. A larger denominator in a fraction results in a smaller value. Thus, \( \frac{1}{9} < \frac{1}{4} \) because 9 is larger than 4.

These comparisons are useful when establishing bounds for sums in the bigger picture of convergence. By comparing fractions, we essentially organize and estimate the behavior of terms in a series, which helps in demonstrating convergence or bounding the sum. For instance, comparing multiple terms to a common denominator allows us to sum and evaluate parts of the series effectively, reinforcing our understanding of where the series converges.

Bounding a series involves defining numerical limits within which the series' sums fall. Establishing bounds ensures the series doesn't grow without limit, showcasing its convergent nature. Essentially, bounding is about framing the sum so it doesn't surpass certain values. In our example, specific parts of the series like \( \frac{1}{4^2} + \frac{1}{5^2} + ... \), contribute less than a particular fraction, ensuring the overall sum remains stable.

Understanding these bounds helps demonstrate the convergence process, as it provides a way to predict the limit towards which the series sums are heading. These estimated bounds offer the theoretical backing needed to substantiate claims about the series' convergence properties and its ultimate sum.

Partial sums are the summation of a select number of initial terms in a series. Think of them as snapshots of the series as it builds towards an infinite sum. These are calculated by adding the terms one by one, starting from the first and moving forward.

In our series, the partial sum calculation allows one to see the gradual growth of the sum. For instance, summing only the first few terms, like \( \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} \), gives a glimpse into how the series begins to shape up and converge.

The concept of partial sums is crucial in proving convergence because it shows how, even as more terms are added, the sum approaches a specific value — providing evidence that the series stabilizes within a predictable range. This methodical approach allows mathematicians to reach conclusions about the series' overall behavior and final sum.