Evaluate the Summation sum from x=0 to 1 of 2x-1
The question is asking you to compute the total value of a mathematical expression that follows a specific rule as you sum it over a certain range. The operation to be performed is summation, which involves adding up all the values calculated by the given expression for each increment within the specified range. The expression provided is "2x-1", and you are expected to evaluate it at every integer point of 'x' starting from 'x=0' and ending at 'x=1'. Essentially, it's asking for the sum of the values of "2x-1" for all the values of 'x' in the range 0 to 1, which includes just the two integer points 0 and 1.
$\sum_{x = 0}^{1} 2 x - 1$
Answer
Solution:
Step 1:
Write out the terms of the series for each value of \( x \): \( 2 \cdot 0 - 1 \) and \( 2 \cdot 1 - 1 \).
Step 2:
Perform the calculations.
Step 2.1:
Calculate \( 2 \times 0 \): \( 0 - 1 + 2 \times 1 - 1 \).
Step 2.2:
Subtract \( 1 \) from the previous result: \( -1 + 2 \times 1 - 1 \).
Step 2.3:
Calculate \( 2 \times 1 \): \( -1 + 2 - 1 \).
Step 2.4:
Subtract \( 1 \) from \( 2 \): \( -1 + 1 \).
Step 2.5:
Combine \( -1 \) and \( 1 \): \( 0 \).
Knowledge Notes:
To evaluate the given summation, we need to understand the concept of a finite series and basic arithmetic operations. A finite series is a sum of terms that are generated by substituting values into a given function, in this case, \( 2x - 1 \), where \( x \) takes on values from a specified range. In this problem, \( x \) ranges from 0 to 1.
The steps involve substituting each value of \( x \) into the function \( 2x - 1 \) and performing the arithmetic operations indicated. This includes multiplication (denoted by \( \cdot \) or \( \times \)) and subtraction (denoted by \( - \)). The summation symbol \( \sum \) indicates that we add the results of the function for each value of \( x \) within the specified range.
The arithmetic operations are performed in a specific order, following the rules of arithmetic, which dictate that multiplication and division are performed before addition and subtraction unless parentheses indicate otherwise. After calculating each term, the final step is to add the results to find the sum of the series. In this case, the series is very short, consisting of only two terms, and the summation is straightforward.
