https://file.aiquickdraw.com/mathbot/file/1725536543549gggnzz8o.jpg

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.

What is the summation of (-1)^(i+1) * i^2 for i=1 to n?

Explore how to compute the summation of (-1)^(i+1) * i^2 for i=1 to n. This problem covers alternating series and square powers, with detailed steps suitable for high school mathematics (Grades 10-12).

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