Evaluate the Summation sum from i=1 to infinity of -4*0.5^(i-1)
The question asks to determine the value of an infinite series, which is a summation of terms in a sequence. Specifically, you are tasked with calculating the sum of terms starting from the index i=1 and continuing indefinitely (to infinity) where each term is given by the expression -4 * 0.5^(i-1). This is a geometric series because each term can be obtained by multiplying the previous one by a fixed ratio, in this case, 0.5. The question requires the application of knowledge on series and convergence of infinite geometric series to evaluate the sum, if it exists.
$\sum_{i = 1}^{\infty} - 4 \cdot \left(0.5\right)^{i - 1}$
Answer
Solution:
Step 1:
To determine the sum of an infinite geometric series, use the formula $S = \frac{a}{1 - r}$, where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio.
Step 2:
Calculate the common ratio $r$ using the formula $r = \frac{a_{i + 1}}{a_{i}}$.
Step 2.1:
Insert $a_{i}$ and $a_{i + 1}$ into the ratio formula: $r = \frac{-4 \cdot (0.5)^{i}}{-4 \cdot (0.5)^{i - 1}}$.
Step 2.2:
Proceed to simplify the expression.
Step 2.2.1:
Eliminate the shared factor of $-4$: $r = \frac{\cancel{-4} \cdot (0.5)^{i}}{\cancel{-4} \cdot (0.5)^{i - 1}}$.
Step 2.2.2:
Reduce the expression by canceling out like terms.
Step 2.2.2.1:
Extract $(0.5)^{i - 1}$ from $(0.5)^{i}$: $r = \frac{(0.5)^{i - 1} \cdot (0.5)^{1}}{(0.5)^{i - 1}}$.
Step 2.2.2.2:
Cancel out identical factors: $r = \frac{\cancel{(0.5)^{i - 1}} \cdot (0.5)^{1}}{\cancel{(0.5)^{i - 1}}}$.
Step 2.2.2.3:
Rewrite the ratio: $r = \frac{(0.5)^{1}}{1}$.
Step 2.2.2.4:
Divide $(0.5)^{1}$ by $1$: $r = (0.5)^{1}$.
Step 2.2.3:
Combine like terms: $r = (0.5)$.
Step 2.2.4:
Simplify the expression: $r = 0.5$.
Step 3:
Confirm that the series converges as $|r| < 1$.
Step 4:
Determine the first term $a$ by substituting $i = 1$ into the term formula: $a = -4 \cdot (0.5)^{0}$.
Step 4.2:
Simplify the first term.
Step 4.2.1:
Evaluate the exponent: $a = -4 \cdot 1$.
Step 4.2.2:
Calculate the first term: $a = -4$.
Step 5:
Insert the values of $a$ and $r$ into the sum formula: $S = \frac{-4}{1 - 0.5}$.
Step 6:
Finalize the simplification.
Step 6.1:
Compute the denominator: $1 - 0.5 = 0.5$.
Step 6.2:
Divide the numerator by the denominator: $S = \frac{-4}{0.5} = -8$.
The sum of the series is $-8$.
Knowledge Notes:
Infinite Geometric Series: A series with an infinite number of terms in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). The series converges if $|r| < 1$.
Sum of an Infinite Geometric Series: The sum can be found using the formula $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio.
Convergence of a Series: A series converges if the absolute value of the common ratio is less than one ($|r| < 1$). If $|r| \geq 1$, the series diverges.
Simplification of Expressions: Involves canceling common factors, reducing fractions, and applying arithmetic operations to arrive at a simpler form.
Exponent Rules: Any number raised to the power of zero equals one ($a^0 = 1$), and multiplying powers with the same base results in adding the exponents ($a^m \cdot a^n = a^{m+n}$).
