91Ó°ÊÓ

A geometric series is a sum of the form \(a + ar + ar^2 + ar^3 + \ldots\). In our exercise, the geometric series is expressed as \( \sum_{n=0}^{\infty}(xy)^{n} \). Here, \(xy\) acts as the common ratio \(r\) for each term in the series. This series is termed 'geometric' because each subsequent term is found by multiplying the previous term by \(r\).

For the series to sum to a finite value, a convergence condition must be satisfied: \(|xy| < 1\). This ensures that the terms get smaller and approach zero as \(n\) increases, allowing the series to sum to a finite value. The sum of an infinite geometric series is calculated using the formula: \( \frac{1}{1-r} \), where \(r = xy\). Thus, the given function can be simplified to \( f(x, y) = \frac{1}{1 - xy} \) under the condition \(|xy| < 1\).

This simplification is crucial as it makes it easier to work with the function, especially when taking derivatives.

The chain rule is a fundamental technique in calculus used for differentiation. It is particularly useful when dealing with composite functions. A composite function is one where the output of one function becomes the input of another. In simpler terms, if you have a function \(g(z)\) and \(z = h(x)\), the chain rule allows you to differentiate \(f(h(x))\) using \(\frac{df}{dx} = \frac{df}{dz} \cdot \frac{dz}{dx}\).

In the solution, the function \(f(x, y) = \frac{1}{1 - xy}\) is treated as a composite function. When we differentiate with respect to \(x\) and \(y\), we apply the chain rule. The differentiation leads to expressions \( \frac{y}{(1 - xy)^2} \) for \(\frac{\partial f}{\partial x}\) and \( \frac{x}{(1 - xy)^2} \) for \(\frac{\partial f}{\partial y}\).

Here's a simple breakdown:

• Differentiating \(1-xy\) inside the function leads \(\frac{d}{dx}(1 - xy) = -y\) and \(\frac{d}{dy}(1 - xy) = -x\).
• Apply this using the chain rule as \(\frac{d}{du} \left( \frac{1}{u} \right) \cdot \frac{du}{dx}\) and \(\frac{du}{dy}\).

This allows us to elegantly find the partial derivatives.

In the realm of infinite series, understanding when a series converges is paramount. Convergence within series representations, such as geometric series, is what ensures that they can be equated to finite sums.

The convergence condition \(|xy| < 1\) is key here. It establishes the parameter within which the series converges, i.e., results in a finite sum. This condition restricts the values \(x\) and \(y\) can assume, ensuring that \(xy\) remains in the interval \(-1, 1\).

Why is this important? The central idea is to avoid divergence, where terms grow indefinitely making it impossible to obtain a finite sum. Convergent series maintain a balance, ensuring terms decrease sufficiently quickly to approach an actual sum. In this specific example, the function \(f(x, y)\) can solely be represented and used in its simplified form \(\frac{1}{1 - xy}\) when the convergence criterion is met.

Understanding such conditions is crucial when dealing with series in calculus and its practical applications.