Sum of 2n series of 1square-2square+3square-4square+5square-6square+....

To tackle the sum of the series \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + \ldots\), we need to observe the pattern in the series and find a method to simplify it. This series alternates between adding and subtracting squares of consecutive integers. Let's break it down step by step.

Understanding the Series Structure

The series can be grouped into pairs of terms for easier analysis. Each pair consists of one positive square and one negative square:

• First pair: \(1^2 - 2^2\)
• Second pair: \(3^2 - 4^2\)
• Third pair: \(5^2 - 6^2\)

We can generalize this pairing. The \(n\)-th pair can be expressed as:

\((2n-1)^2 - (2n)^2\)

Calculating Each Pair

Now let's simplify the expression for each pair:

\((2n-1)^2 - (2n)^2 = (4n^2 - 4n + 1) - (4n^2) = -4n + 1\)

This shows that each pair contributes \(-4n + 1\) to the overall sum of the series.

Finding the Total Sum

Next, we need to determine how many pairs there are in the sequence. If we consider the series up to the \(2n\)-th term, we have \(n\) pairs. Therefore, the entire sum can be computed as:

\(S_n = \sum_{k=1}^{n} (-4k + 1)\)

This can be split into two separate sums:

\(S_n = \sum_{k=1}^{n} -4k + \sum_{k=1}^{n} 1\)

Calculating Each Component

• The first sum, \(\sum_{k=1}^{n} -4k\), is \(-4 \cdot \frac{n(n+1)}{2} = -2n(n+1)\).
• The second sum, \(\sum_{k=1}^{n} 1\), simply counts the number of pairs, which is \(n\).

Combining these results gives:

\(S_n = -2n(n+1) + n\)

Simplifying the Final Expression

Now, let’s simplify:

\(S_n = -2n^2 - 2n + n = -2n^2 - n\)

This means that the sum of the series \(1^2 - 2^2 + 3^2 - 4^2 + 5^2 - 6^2 + \ldots\) up to \(2n\) terms is:

\(S_n = -2n^2 - n\)

Conclusion and Application

So, when you're asked to find the sum of this alternating series, you can use this method of pairing and simplifying to arrive at the answer efficiently. This technique of recognizing patterns and breaking complex problems into manageable parts is a valuable skill in mathematics.