Evaluate the Summation sum from n=1 to infinity of 2(3/2)^(n-1)
The question is asking for the evaluation of an infinite series, specifically finding the sum of the series that includes terms that follow a pattern. The pattern is given by the formula 2(3/2)^(n-1), where n starts at 1 and goes to infinity. The series is therefore a geometric series because each term is a constant multiple of the previous term. You are being asked to calculate the total sum of this infinite geometric series.
$\sum_{n = 1}^{\infty} 2 \left(\left(\right. \frac{3}{2} \left.\right)\right)^{n - 1}$
Answer
Solution:
Step 1:
To determine the sum of an infinite geometric series, use the formula $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio.
Step 2:
Calculate the common ratio $r$ by using the formula $r = \frac{a_{n+1}}{a_n}$.
Step 2.1:
Insert the values of $a_{n}$ and $a_{n+1}$ into the ratio formula: $r = \frac{2(\frac{3}{2})^{n}}{2(\frac{3}{2})^{n-1}}$.
Step 2.2:
Proceed to simplify the ratio.
Step 2.2.1:
Eliminate the factor of $2$ that appears in both the numerator and denominator: $r = \frac{\cancel{2}(\frac{3}{2})^{n}}{\cancel{2}(\frac{3}{2})^{n-1}}$.
Step 2.2.1.1:
The expression simplifies to: $r = \frac{(\frac{3}{2})^{n}}{(\frac{3}{2})^{n-1}}$.
Step 2.2.2:
Cancel out the common base of $(\frac{3}{2})^{n-1}$ in both the numerator and denominator.
Step 2.2.2.1:
Factor out $(\frac{3}{2})^{n-1}$ from $(\frac{3}{2})^{n}$: $r = \frac{(\frac{3}{2})^{n-1}(\frac{3}{2})^{1}}{(\frac{3}{2})^{n-1}}$.
Step 2.2.2.2:
Eliminate the common factors.
Step 2.2.2.2.1:
Introduce a multiplication by $1$: $r = \frac{(\frac{3}{2})^{n-1}(\frac{3}{2})^{1}}{(\frac{3}{2})^{n-1} \cdot 1}$.
Step 2.2.2.2.2:
Cancel out the common base: $r = \frac{\cancel{(\frac{3}{2})^{n-1}}(\frac{3}{2})^{1}}{\cancel{(\frac{3}{2})^{n-1}} \cdot 1}$.
Step 2.2.2.2.3:
The ratio simplifies to: $r = \frac{(\frac{3}{2})^{1}}{1}$.
Step 2.2.2.2.4:
Divide by $1$: $r = (\frac{3}{2})^{1}$.
Step 2.2.3:
Combine the terms: $r = (\frac{3}{2})^{1 - (n - n)}$.
Step 2.2.4:
Simplify the exponent.
Step 2.2.4.1:
Apply the distributive property: $r = (\frac{3}{2})^{1 - n + n}$.
Step 2.2.4.2:
Simplify the expression: $r = (\frac{3}{2})^{1}$.
Step 2.2.5:
Subtract $n$ from $n$: $r = (\frac{3}{2})^{1}$.
Step 2.2.6:
Add the remaining terms: $r = (\frac{3}{2})^{1}$.
Step 2.2.7:
The final value of $r$ is: $r = \frac{3}{2}$.
Step 3:
Determine if the series converges by checking if $|r| < 1$. Since $|r| = \frac{3}{2} > 1$, the series does not converge.
Knowledge Notes:
Infinite Geometric Series: An infinite geometric series is a series with a constant ratio between successive terms. It can be represented as $a + ar + ar^2 + ar^3 + \ldots$, where $a$ is the first term and $r$ is the common ratio.
Sum of an Infinite Geometric Series: The sum of an infinite geometric series can be found using the formula $S = \frac{a}{1 - r}$, provided that $|r| < 1$. If $|r| \geq 1$, the series does not converge to a finite sum.
Common Ratio: The common ratio $r$ in a geometric series is the factor by which each term is multiplied to get the next term. It is calculated as $r = \frac{a_{n+1}}{a_n}$.
Convergence Criteria: For an infinite geometric series to converge, the absolute value of the common ratio must be less than one ($|r| < 1$). If the absolute value of the common ratio is greater than or equal to one, the series diverges.
Simplification of Expressions: When simplifying expressions, common factors in the numerator and denominator can be canceled out. Additionally, properties of exponents can be used to combine or separate terms with the same base.
Absolute Value: The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. For a real number $x$, the absolute value is denoted as $|x|$.
