Evaluate the Summation sum from j=1 to 5 of (-1)^jj

Solution:

Step 1: Expansion of the Summation

Write out the terms of the summation for $j$ ranging from 1 to 5:

$(-1)^1 \cdot 1 + (-1)^2 \cdot 2 + (-1)^3 \cdot 3 + (-1)^4 \cdot 4 + (-1)^5 \cdot 5$

Step 2: Simplification of the Series

Evaluate each term and then add them together:

$-1 + 2 - 3 + 4 - 5 = -3$

Knowledge Notes:

The problem involves evaluating a finite summation where each term is given by $(-1)^j \cdot j$. This is a series where the sign of each term alternates due to the $(-1)^j$ factor, and the magnitude of each term increases linearly with $j$.

To solve this problem, we follow these steps:

1. Expansion of the Summation: We list out each term of the series by substituting the values of $j$ from 1 to 5 into the expression $(-1)^j \cdot j$. This gives us a clear view of the series and helps us see the pattern of alternating signs.

2. Simplification of the Series: After expanding the series, we simplify it by performing the exponentiation and multiplication for each term. The exponentiation of $(-1)$ to an integer power results in $-1$ if the power is odd and $1$ if the power is even. This causes the alternating sign pattern. After calculating each term, we add them together to find the sum of the series.

Relevant mathematical concepts include:

• Exponentiation: When a number is raised to a power, it is multiplied by itself as many times as the value of the power. In this case, $(-1)$ raised to any power determines the sign of the term.

• Summation: The process of adding a sequence of numbers; the result is their sum or total.

• Series: A summation of a sequence of terms that follow a certain pattern or rule.

• Arithmetic Progression: A sequence of numbers in which each term after the first is obtained by adding a constant difference to the previous term. However, in this problem, the sequence is not a simple arithmetic progression due to the alternating sign.