13. Explain what is wrong with this solution. $$ \begin{array}{l}1+2+4+8 \quad ext { is an infnite geometric series with } a=1 ext { and } r=2 ext {. } \ \$=\frac{1}{1}-2=-1 \ 1+2+4+8+\ldots=-1 \ ext { So by adding infinitely many positive numbers we get a negative answeri }\end{array} $$

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13. Explain what is wrong with this solution. $$ \begin{array}{l}1+2+4+8 \quad 	ext { is an infnite geometric series with } a=1 	ext { and } r=2 	ext {. } \ \$=\frac{1}{1}-2=-1 \ 1+2+4+8+\ldots=-1 \ 	ext { So by adding infinitely many positive numbers we get a negative answeri }\end{array} $$

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The solution presented contains a fundamental misunderstanding of the conditions under which the formula for the sum of an infinite geometric series is valid. Here's a detailed breakdown of the error: ### **1. Misapplication of the Geometric Series Formula** The formula used to find the sum $$ S $$ of an infinite geometric series is: $$ S = \frac{a}{1 - r} $$ **However, this formula is only valid when the absolute value of the common ratio satisfies $$ |r| < 1 $$.** This condition ensures that the terms of the series decrease in magnitude and approach zero, allowing the series to converge to a finite sum. ### **2. Incorrect Parameters** In the given series: $$ 1 + 2 + 4 + 8 + \ldots $$ - The **first term** $$ a = 1 $$ - The **common ratio** $$ r = 2 $$ Here, $$ r = 2 $$ clearly does **not** satisfy $$ |r| < 1 $$; instead, $$ |r| > 1 $$. ### **3. Consequences of $$ |r| \geq 1 $$** When $$ |r| \geq 1 $$: - **If $$ r > 1 $$**: Each subsequent term is larger than the previous one, causing the series to grow without bound. - **If $$ r = 1 $$**: All terms are equal, and the series sums to infinity since you're continuously adding the same positive number. - **If $$ r < -1 $$**: The terms oscillate in sign and grow in magnitude, preventing the series from settling to a finite sum. In the specific case of $$ r = 2 $$: $$ 1 + 2 + 4 + 8 + \ldots \rightarrow \infty $$ The series **diverges**, meaning it does not approach any finite value. ### **4. Logical Inconsistency** The erroneous conclusion that an infinite sum of positive numbers equals a negative number ($$-1$$) is illogical within the standard framework of real numbers. This contradiction arises because the formula was applied outside its valid range, leading to an invalid result. ### **Conclusion** The main mistake is using the infinite geometric series sum formula $$ S = \frac{a}{1 - r} $$ without verifying that the common ratio $$ r $$ satisfies $$ |r| < 1 $$. Since $$ r = 2 $$ in this case does not meet this condition, the series does not have a finite sum, and the application of the formula is invalid. **Final Answer:** They incorrectly applied the infinite geometric series formula without checking that r = 2 exceeds 1. The formula S = a/(1 – r) only holds when |r| < 1, so the series actually diverges and does not sum to –1. They used the wrong formula for the sum of an infinite geometric series because the common ratio $$ r = 2 $$ is greater than 1, which is not allowed. The series actually diverges and does not have a finite sum.

13. Explain what is wrong with this solution. \( \begin{array}{l}1+2+4+8 \quad \text { is an infnite geometric series with } a=1 \text { and } r=2 \text {. } \\ \$=\frac{1}{1}-2=-1 \\ 1+2+4+8+\ldots=-1 \\ \text { So by adding infinitely many positive numbers we get a negative answeri }\end{array} \)

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