Evaluate the Summation sum from k=1 to 4 of 1/(k-k^2)
The question asks for the evaluation of a mathematical expression that involves a summation (also known as a sigma notation). Specifically, it requires calculating the sum of the terms 1/(k - k^2) for each integer value of k starting at 1 and going up to 4. The summation involves adding the results of the expression within the sum for each value of k, which results in a single numerical answer. The expression to be summed is a rational function where k appears both in the numerator and the denominator in a quadratic form. This summation requires understanding of algebraic manipulation and potentially the properties of series or fractions.
$\sum_{k = 1}^{4} \frac{1}{k - k^{2}}$
Answer
Solution:
Step:1
Decompose the expression within the summation.
Step:1.1
Extract the common factor $k$ from the expression $k - k^{2}$.
Step:1.1.1
Elevate $k$ to the first power to get $\frac{1}{k^{1} - k^{2}}$.
Step:1.1.2
Extract the factor $k$ from $k^{1}$ to obtain $\frac{1}{k \cdot 1 - k^{2}}$.
Step:1.1.3
Extract the factor $k$ from $-k^{2}$ to get $\frac{1}{k \cdot 1 + k(-k)}$.
Step:1.1.4
Remove the common factor $k$ from $k \cdot 1 + k(-k)$ to simplify to $\frac{1}{k(1 - k)}$.
Step:1.2
Reformulate the summation as $\sum_{k = 1}^{4} \frac{1}{k(1 - k)}$.
Step:2
Calculate the series by substituting each value of $k$ into the expression.
$\frac{1}{1(1 - 1)} + \frac{1}{2(1 - 2)} + \frac{1}{3(1 - 3)} + \frac{1}{4(1 - 4)}$
Step:3
Simplify each term in the series.
Step:3.1
Evaluate $1 - 1$ within the first term to get $\frac{1}{1(1 - 1)} + \frac{1}{2(1 - 2)} + \frac{1}{3(1 - 3)} + \frac{1}{4(1 - 4)}$.
Step:3.2
Calculate $-1 \cdot 1$ to continue simplification $\frac{1}{1 - 1} + \frac{1}{2(1 - 2)} + \frac{1}{3(1 - 3)} + \frac{1}{4(1 - 4)}$.
Step:3.3
Subtract $1$ from $1$ to further simplify $\frac{1}{0} + \frac{1}{2(1 - 2)} + \frac{1}{3(1 - 3)} + \frac{1}{4(1 - 4)}$.
Step:3.4
Identify the division by zero in the expression, which renders the expression undefined.
The final result is Undefined.
Knowledge Notes:
The problem involves evaluating a summation of a rational function where the denominator is a quadratic expression in terms of $k$. The process includes the following knowledge points:
Factoring: The ability to factor expressions is crucial in simplifying algebraic fractions. In this case, factoring out $k$ from the denominator simplifies the expression.
Exponents: Understanding that $k$ can be written as $k^1$ is important for factoring.
Summation: Recognizing how to write and evaluate a summation expression is key to solving this problem. The summation symbol $\sum$ indicates that we need to evaluate the expression for each integer value of $k$ from 1 to 4 and sum the results.
Undefined Expressions: Division by zero is undefined in mathematics. If at any point in the evaluation a division by zero occurs, the entire expression becomes undefined.
Simplification: The process of simplifying algebraic expressions involves performing operations like multiplication and subtraction to reduce the expression to its simplest form.
In this problem, the summation cannot be evaluated because the first term in the series results in a division by zero, which is undefined. This is a critical concept in mathematics, as undefined expressions do not have a value in the real number system.
