Evaluate the Summation sum from j=0 to 4 of (-1)^j
The given problem is a mathematical question asking to compute the value of a specific summation. The summation is defined by the sum of terms from j=0 to j=4, and each term in the summation is given by (-1) raised to the power of j. The caret symbol (^) denotes exponentiation. The value of j runs through the integer values starting from 0 up to and including 4. For each value of j, you would calculate (-1)^j, which would alternate between 1 and -1 because (-1) to an even power is 1, and (-1) to an odd power is -1. The task is to add up these individual terms to find the total sum.
$\sum_{j = 0}^{4} \left(\left(\right. - 1 \left.\right)\right)^{j}$
Answer
Solution:
Step 1: Write out the terms of the series
The series is expanded to show each term for $j$ ranging from 0 to 4:
$(-1)^0 + (-1)^1 + (-1)^2 + (-1)^3 + (-1)^4$
Step 2: Calculate the value of each term and sum them
Evaluate each term and add them together:
$1 - 1 + 1 - 1 + 1 = 1$
Knowledge Notes:
The problem involves evaluating a finite summation of terms of the form $(-1)^j$. This is a geometric series where the common ratio is $-1$. In a geometric series, each term is the product of the previous term and a constant multiplier (the common ratio). The series alternates between positive and negative terms because the base of the exponent is $-1$.
Key points to understand in solving this problem:
Exponents of Negative Numbers: When a negative number is raised to an even exponent, the result is positive. When it is raised to an odd exponent, the result is negative.
Geometric Series: A series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ratio.
Summation Notation: The summation symbol $\Sigma$ represents the sum of a sequence of numbers, according to a given rule. In this case, the rule is the sequence of terms $(-1)^j$ for $j$ from 0 to 4.
Finite Series: A series that has a limited number of terms. The summation in this problem is finite because it sums over the five terms from $j=0$ to $j=4$.
The solution involves two steps: expanding the series to show each term and then simplifying the series by calculating the value of each term and summing them. The alternating signs result in a pattern where positive and negative terms cancel each other out, leading to the final simplified result.
