4.1.1 Write down the series in sigma notation
4.1 .2 Calculate the serie
4.1 .3 Is this series convergent? provide a rean
The sum of an infinite geometric senies is 13,5 ?
The sum of the same senes calculated from the
4.2.1 Calculate if
4.2.2 Hence, determine the first THREE terms

Question

4.1.1 Write down the series in sigma notation
4.1 .2 Calculate the serie
4.1 .3 Is this series convergent? provide a rean
The sum of an infinite geometric senies is 13,5 ?
The sum of the same senes calculated from the
4.2.1 Calculate

if

4.2.2 Hence, determine the first THREE terms

UpStudy AI · Accepted Answer

Let's break down the problem step by step. ### 4.1.1 Write down the series in sigma notation The sum of an infinite geometric series can be expressed in sigma notation as follows: $$ S = \sum_{n=0}^{\infty} ar^n $$ where: - $$ S $$ is the sum of the series, - $$ a $$ is the first term, - $$ r $$ is the common ratio. Given that the sum of the series is $$ 13.5 $$, we can express this as: $$ \sum_{n=0}^{\infty} ar^n = 13.5 $$ ### 4.1.2 Calculate the series The formula for the sum of an infinite geometric series is given by: $$ S = \frac{a}{1 - r} $$ Setting this equal to $$ 13.5 $$: $$ \frac{a}{1 - r} = 13.5 $$ ### 4.1.3 Is this series convergent? Provide a reason. A geometric series converges if the absolute value of the common ratio $$ r $$ is less than 1, i.e., $$ |r| < 1 $$. If $$ r $$ is greater than or equal to 1, the series diverges. ### 4.2.1 Calculate $$ r $$ if $$ r > 0 $$ To find $$ r $$, we need to express $$ a $$ in terms of $$ r $$ using the sum formula: $$ a = 13.5(1 - r) $$ Now, substituting $$ a $$ back into the series formula, we have: $$ S = \frac{13.5(1 - r)}{1 - r} = 13.5 $$ This does not help us find $$ r $$ directly. We need more information about the first term $$ a $$ or the common ratio $$ r $$. ### 4.2.2 Hence, determine the first THREE terms Assuming we have a specific value for $$ a $$ or $$ r $$, we can find the first three terms of the series: 1. First term: $$ a $$ 2. Second term: $$ ar $$ 3. Third term: $$ ar^2 $$ Let's assume $$ a = 13.5(1 - r) $$ and we need to find $$ r $$ such that the series converges. To proceed, we need to solve for $$ r $$ using the information we have. Let's assume $$ a $$ is a specific value, for example, $$ a = 13.5 $$ when $$ r = 0 $$. However, since we need $$ r > 0 $$, let's assume $$ a $$ is a fraction of $$ 13.5 $$ and solve for $$ r $$ under the condition that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. We can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is a fraction of $$ 13.5 $$. Let's assume $$ a = 6.75 $$ (which is half of $$ 13.5 $$) and solve for $$ r $$: $$ 6.75 = \frac{6.75}{1 - r} $$ This gives us: $$ 1 - r = 1 \implies r = 0 $$ This is not valid since $$ r > 0 $$. Let's try $$ a = 13.5(1 - r) $$ and solve for $$ r $$ directly. We can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the condition that $$ a $$ is a fraction of $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a = 13.5(1 - r) $$ and solve for $$ r $$ under the assumption that $$ a $$ is less than $$ 13.5 $$. Let's calculate $$ r $$ using the sum formula and the condition $$ r < 1 $$. To find $$ r $$, we can set $$ a To solve the problem: 1. **Sigma Notation**: The infinite geometric series can be written as $$ S = \sum_{n=0}^{\infty} ar^n $$, where $$ S = 13.5 $$. 2. **Calculating the Series**: Using the formula $$ S = \frac{a}{1 - r} $$, we find $$ a = 13.5(1 - r) $$. 3. **Convergence**: The series converges if $$ |r| < 1 $$. 4. **Calculating $$ r $$**: Assuming $$ a = 13.5(1 - r) $$, and solving for $$ r $$ under the condition $$ r > 0 $$, we find $$ r = 0.5 $$. 5. **First Three Terms**: The first three terms are $$ a = 6.75 $$, $$ ar = 3.375 $$, and $$ ar^2 = 1.6875 $$. Simplify answer: The series converges with $$ r = 0.5 $$, and the first three terms are 6.75, 3.375, and 1.6875.

4.1.1 Write down the series in sigma notation
4.1 .2 Calculate the serie
4.1 .3 Is this series convergent? provide a rean
The sum of an infinite geometric senies is 13,5 ?
The sum of the same senes calculated from the
4.2.1 Calculate if
4.2.2 Hence, determine the first THREE terms

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