Evaluate the Summation sum from j=1 to 6 of (-1)^jj
The problem asks to calculate the result of a finite summation of terms, where each term is defined by the formula (-1)^j * j. The summation is to be evaluated from j=1 to j=6. This involves taking each integer value of j from 1 to 6, substituting it into the formula to calculate each term, multiplying that by a -1 raised to the power of j (which determines whether the term is positive or negative), and finally summing up all the resulting terms to get the total sum.
$\sum_{j = 1}^{6} \left(\left(\right. - 1 \left.\right)\right)^{j} j$
Answer
Solution:
Evaluate the Summation
Step 1: Expand the Series
Write out each term of the series by substituting $j$ with values from 1 to 6.
$(-1)^1 \cdot 1 + (-1)^2 \cdot 2 + (-1)^3 \cdot 3 + \ldots + (-1)^6 \cdot 6$
Step 2: Simplify the Series
Calculate the sum of the expanded series.
$3$
Knowledge Notes:
To solve the given problem, we need to understand the concept of summation and the properties of exponents.
Summation: The summation symbol $\Sigma$ represents the sum of a sequence of numbers. The expression $\sum_{j=1}^{n} a_j$ means to sum the sequence $a_j$ from $j=1$ to $j=n$.
Exponents: The exponentiation of a number tells us how many times to multiply the number by itself. The expression $(-1)^j$ means to multiply -1 by itself $j$ times. If $j$ is even, $(-1)^j$ will be 1, and if $j$ is odd, $(-1)^j$ will be -1.
Combining Summation and Exponents: In the given problem, we are asked to evaluate the sum of $(-1)^j \cdot j$ from $j=1$ to $j=6$. This requires us to consider the sign change due to the exponent on -1 and multiply it by the current value of $j$.
Simplifying the Series: After expanding the series, we can simplify it by calculating the value of each term and then summing them up. Since the power of -1 alternates between positive and negative values, the series will have alternating positive and negative terms.
In LaTeX, we use the $\sum$command to denote summation, and we use $^$to denote exponents. To write the series in LaTeX, we would use the expression:
$$\sum_{j=1}^{6} (-1)^j \cdot j$$
To expand this series, we substitute $j$ with each number from 1 to 6 and calculate the value of each term. The simplified result of this summation is the final answer to the problem.
