Evaluate the Summation sum from j=1 to 8 of (-1)^jj
The problem is asking to calculate the total value of a mathematical series where the summation iterates from j=1 to j=8. In this series, each term consists of the integer j multiplied by the value (-1) raised to the power j. The notation (-1)^j means that -1 is the base and j is the exponent, resulting in alternating signs for consecutive terms. So, the summation will involve adding and subtracting subsequent values of j, depending on whether j is odd (which would give a negative contribution) or even (which would give a positive contribution). The series adds up these terms in a specific sequence to find the final sum.
$\sum_{j = 1}^{8} \left(\left(\right. - 1 \left.\right)\right)^{j} j$
Answer
Solution:
Step 1:
Write out the terms of the summation for each integer $j$ from 1 to 8.
$(-1)^1 \cdot 1 + (-1)^2 \cdot 2 + (-1)^3 \cdot 3 + \ldots + (-1)^8 \cdot 8$
Step 2:
Evaluate the simplified form of the series.
$4$
Knowledge Notes:
The problem involves evaluating a finite summation, which is a common operation in mathematics where you add up a sequence of terms generated by a formula that changes with each term. The summation in question is:
$$\sum_{j=1}^{8} (-1)^j j$$
This summation has two parts: $(-1)^j$ and $j$. The $(-1)^j$ part alternates the sign of each term in the sequence, while $j$ represents the value of each term in the sequence.
Here are some relevant knowledge points related to this problem:
Summation Notation: Summation notation is a way to represent the addition of a sequence of numbers. The symbol $\sum$ is used, followed by an expression that describes the terms to be added. The variable below the $\sum$ symbol is the index of summation, and the numbers below and above the symbol indicate the starting and ending values, respectively.
Exponents: An exponent indicates how many times a number, known as the base, is multiplied by itself. In this problem, $(-1)^j$ is the base raised to the power of $j$. When the base is negative and the exponent is even, the result is positive; when the exponent is odd, the result is negative.
Arithmetic Series: An arithmetic series is 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 series is not purely arithmetic because of the alternating sign.
Series Expansion: Expanding a series means writing out all the terms of the series explicitly. This can be helpful for visualizing the pattern of the series and for performing calculations on a term-by-term basis.
Simplification: Simplifying an expression involves combining like terms and reducing the expression to its simplest form. In the context of series, this might involve adding and subtracting the terms that have been expanded.
In this particular problem, the series alternates between positive and negative terms, which is characteristic of an alternating series. When expanded, the terms with even exponents will be positive, and the terms with odd exponents will be negative. After expanding and simplifying, the result is a single number, which in this case is 4.
