Evaluate the Summation sum from n=1 to 8 of (-1)^n

Solution:

Step 1:

To calculate the sum of a finite geometric series, use the formula $S=a\left(\frac{1-r^n}{1-r}\right)$, where $a$ is the initial term and $r$ is the common ratio.

Step 2:

Determine the common ratio $r$ by using the formula $r=\frac{a_{n+1}}{a_n}$.

Step 2.1:

Insert $a_n$ and $a_{n+1}$ into the ratio formula: $r=\frac{(-1{)}^{n+1}}{(-1{)}^n}$.

Step 2.2:

Eliminate the similar factor from $(-1{)}^{n+1}$ and $(-1{)}^n$.

Step 2.2.1:

Extract $(-1{)}^n$ from $(-1{)}^{n+1}$: $r=\frac{(-1{)}^n\cdot (-1)}{(-1{)}^n}$.

Step 2.2.2:

Remove the common factors.

Step 2.2.2.1:

Multiply by unity: $r=\frac{(-1{)}^n\cdot (-1)}{(-1{)}^n\cdot 1}$.

Step 2.2.2.2:

Cancel out the common factor: $r=\frac{\cancel{(-1{)}^n}\cdot (-1)}{\cancel{(-1{)}^n}\cdot 1}$.

Step 2.2.2.3:

Reformulate the expression: $r=\frac{-1}{1}$.

Step 2.2.2.4:

Divide $-1$ by $1$: $r=-1$.

Step 3:

Identify the first term by substituting the lower limit into $(-1{)}^n$.

Step 3.1:

Replace $n$ with $1$: $a=(-1{)}^1$.

Step 3.2:

Compute the exponent: $a=-1$.

Step 4:

Plug in the values for $r$, $a$, and the number of terms into the summation formula: $S=-\frac{1-(-1{)}^8}{1-(-1)}$.

Step 5:

Simplify the expression.

Step 5.1:

Condense the numerator.

Step 5.1.1:

Combine $-1$ with $(-1{)}^8$ by summing the exponents.

Step 5.1.1.1:

Multiply $-1$ by $(-1{)}^8$.

Step 5.1.1.1.1:

Raise $-1$ to the first power: $S=-\frac{1+(-1{)}^1\cdot (-1{)}^8}{1-(-1)}$.

Step 5.1.1.1.2:

Apply the exponent rule $a^m\cdot a^n=a^{m+n}$: $S=-\frac{1+(-1{)}^{1+8}}{1-(-1)}$.

Step 5.1.1.2:

Sum the exponents: $S=-\frac{1+(-1{)}^9}{1-(-1)}$.

Step 5.1.2:

Elevate $-1$ to the ninth power: $S=-\frac{1-1}{1-(-1)}$.

Step 5.1.3:

Subtract one from one: $S=-\frac{0}{1-(-1)}$.

Step 5.2:

Streamline the denominator.

Step 5.2.1:

Multiply $-1$ by $-1$: $S=-\frac{0}{1+1}$.

Step 5.2.2:

Add one to one: $S=-\frac{0}{2}$.

Step 5.3:

Divide zero by two: $S=-0$.

Step 5.4:

Multiply $-1$ by zero: $S=0$.

Knowledge Notes:

• Geometric Series: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). The sum of the first $n$ terms of a geometric series can be calculated using the formula $S=a\left(\frac{1-r^n}{1-r}\right)$, where $a$ is the first term and $n$ is the number of terms.

• Common Ratio: The common ratio in a geometric series is the constant factor between consecutive terms. It is calculated by dividing any term by the preceding term: $r=\frac{a_{n+1}}{a_n}$.

• Exponent Rules: When multiplying powers with the same base, you can add the exponents: $a^m\cdot a^n=a^{m+n}$. When raising a power to a power, you multiply the exponents: $(a^m{)}^n=a^{m\cdot n}$.

• Negative Exponents: A negative exponent indicates that the base is on the wrong side of a fraction and needs to be inverted. For example, $a^{-n}=\frac{1}{a^n}$.

• Simplification: The process of simplifying an expression involves combining like terms, reducing fractions, and performing arithmetic operations to achieve the simplest form of the expression.

• Finite Geometric Series: The sum of a finite geometric series is found using the formula provided, and it only works when the absolute value of the common ratio is less than one ($|r|<1$). In this problem, the series is finite and has a common ratio of $-1$.

• Even and Odd Powers: Raising $-1$ to an even power results in $1$, while raising it to an odd power results in $-1$. This is due to the fact that an even number of negative factors will always multiply to a positive result, and an odd number will result in a negative.