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The comparison test is a fundamental tool in calculus for determining whether a series converges. It involves comparing a given series with another series that is already known to converge or diverge. Here's how it works:

• If you have two series, \(\textstyle \bigsum a_n\) and \(\textstyle \bigsum b_n\), and \(\textstyle a_n \leq b_n\) for all n, then:
• If \(\textstyle \bigsum b_n\) converges, so does \(\textstyle \bigsum a_n\).
• If \(\textstyle \bigsum a_n\) diverges, so does \(\textstyle \bigsum b_n\).

In the given exercise, the comparison test is applied to \(\textstyle \bigsum \sqrt{a_n b_n}\) which has been shown to be less than or equal to \(\textstyle \bigsum (a_n + b_n)\). Since both \(\textstyle \bigsum a_n\) and \(\textstyle \bigsum b_n\) are convergent series, their sum \(\textstyle \bigsum (a_n + b_n)\) is also convergent. Hence, by the comparison test, \(\textstyle \bigsum \sqrt{a_n b_n}\) must also converge.

A non-negative series is a series where each term is zero or positive. This property simplifies convergence analysis since we don’t have to deal with potential cancellation effects due to negative terms.

• For example, given series \(\textstyle \bigsum a_n\) and \(\textstyle \bigsum b_n\) with \(\textstyle a_n \geq 0\) and \(\textstyle b_n \geq 0\) for all n, ensures the summation results in non-negative numbers.

The significance of non-negative terms in this problem is critical. They allow us to employ tools like the comparison test without worrying about negative values affecting the convergence. In this exercise:

• Given \(\textstyle \bigsum a_n\) and \(\textstyle \bigsum b_n\) are convergent non-negative series, we are assured that manipulating these terms using inequalities like \(\textstyle \sqrt{a_n b_n} \leq a_n + b_n\) won’t encounter issues with negative values.

In mathematics, proving inequalities is crucial as it establishes bounds and relationships between different expressions. In this exercise, the inequality \(\textstyle \sqrt{a_n b_n} \leq a_n + b_n\) plays a vital role. Here is the step-by-step breakdown of proving it:

• Start with \(\textstyle \sqrt{a_n b_n} \leq a_n + b_n\). Square both sides to simplify comparison of non-linear terms.

This gives: \(\textstyle a_n b_n \leq (a_n + b_n)^2\).

• Next, expand the right side: \(\textstyle a_n b_n \leq a_n^2 + 2a_n b_n + b_n^2\).
• Subtract \(\textstyle a_n b_n\) from both sides to obtain : \(\textstyle 0 \leq a_n^2 + a_n b_n + b_n^2\).

Because all the terms \(\textstyle a_n^2\), \(\textstyle a_n b_n\), and \(\textstyle b_n^2\) are non-negative, this inequality holds true for all non-negative \(\textstyle a_n\) and \(\textstyle b_n\).
This inequality is crucial as it allows us to set a boundary for \(\textstyle \sqrt{a_n b_n}\) in terms of \(\textstyle a_n\) and \(\textstyle b_n\), leading to the application of the comparison test.

Proof techniques in calculus often involve a mix of algebra, inequalities, and logical reasoning. In the given exercise, multiple proof techniques come together to demonstrate the convergence of \(\textstyle \bigsum \sqrt{a_n b_n}\). The main techniques used include:

• **Direct comparison:** Showing \(\textstyle \sqrt{a_n b_n}\) is bounded by a known convergent series.

• **Inequality manipulation:** Squaring and expanding terms to simplify the expression and identify how it can be bounded.

• **Summation properties:** Using properties of summations such as the combination of convergent series, i.e., if \(\textstyle \bigsum a_n\) and \(\textstyle \bigsum b_n\) converge, then \(\textstyle \bigsum (a_n + b_n)\) also converges.

Combining these techniques provides a structured approach to solving the problem. Calculus proof techniques are essential for analyzing and concluding the behavior of functions and series mathematically.