Let's try to solve the sum

$S = \sum \left( \frac{n^2 + n + 1}{n^2 - n + 1} \right) (3n^2 - 1)$

using Cauchy's method. Cauchy's method is often used for handling sums by simplifying them into manageable forms and examining series behavior. Here's the step-by-step breakdown:

1. Expression Expansion:

We begin by expanding the terms inside the summation.

$\frac{n^2 + n + 1}{n^2 - n + 1}(3n^2 - 1)$

This involves multiplying the rational expression by $3n^2 - 1$:

$= \frac{(n^2 + n + 1)(3n^2 - 1)}{n^2 - n + 1}$

Now, expand the numerator:

$(n^2 + n + 1)(3n^2 - 1) = n^2(3n^2 - 1) + n(3n^2 - 1) + 1(3n^2 - 1)$

Breaking this down:

$= 3n^4 - n^2 + 3n^3 - n + 3n^2 - 1$

Simplifying:

$= 3n^4 + 3n^3 + 2n^2 - n - 1$

So the entire expression now looks like:

$S = \sum \frac{3n^4 + 3n^3 + 2n^2 - n - 1}{n^2 - n + 1}$

2. Simplify Using Cauchy’s Method:

The Cauchy method involves analyzing the series for specific properties. We look for patterns in the behavior of terms. For simplicity, you can examine whether any of the terms simplify as a result of the form of the denominator $n^2 - n + 1$. Notice that this is a quadratic form, and we want to check if the numerator can be simplified further in a useful way.

Since this sum is non-trivial, a direct analytical solution will require careful manipulation, and depending on the exact context or limits of the sum (e.g., if it’s infinite or finite, and over which values of $n$), more specific steps can be employed to find the value of $S$.

However, this approach gives us an idea of the complexity of the series and how breaking it into individual polynomial terms can aid in evaluation.

Would you like to proceed with specific limits or conditions, such as calculating for finite sums, or would you like to continue breaking down the terms further?

Let me know if you'd like further details!

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Some related questions:

1. How do you handle infinite series using Cauchy’s method?
2. Can we simplify this sum further by recognizing any known series patterns?
3. What is the difference between a Cauchy method approach and a telescoping sum?
4. Can we use Cauchy's method to evaluate sums of trigonometric or exponential functions?
5. How do you handle sums with higher powers, like $n^4$, in Cauchy’s method?

Tip: In Cauchy’s method, try to express the sum as a simplified series or transform it into a known summation form (like a geometric or telescoping series) for easier evaluation!