The image shows the following summation:

$\sum_{i=1}^{n} (-1)^{i+1} \cdot i^2$

This is a finite series where each term alternates in sign based on $(-1)^{i+1}$, and each term is the square of $i$. Let's break it down:

• The term $(-1)^{i+1}$ ensures alternating signs.
• The term $i^2$ is simply the square of the index $i$.

General Form:

For $i = 1$ to $n$: $S = (-1)^{2} \cdot 1^2 + (-1)^{3} \cdot 2^2 + (-1)^{4} \cdot 3^2 + \dots + (-1)^{n+1} \cdot n^2$ This series alternates between positive and negative squares of integers.

Do you want further details on how to compute or analyze this summation for a specific $n$? Let me know!

Here are some related questions:

1. What is the closed form of this summation for specific $n$?
2. How can we generalize this series to non-integer exponents?
3. What are the properties of alternating series, and when do they converge?
4. How can this be related to alternating power series expansions?
5. What real-world problems might this summation model?

Tip: Alternating series can often be simplified by grouping terms based on symmetry or cancellation patterns.