Question
Compute each sum below. Give exact values, not decimal approximations. If the sum does not exist, click on "No sum". \[ 5+5(3)+5(3)^{2}+\ldots=\square \]
Ask by Matthews Gross. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The series does not have a sum.
Solution
1. Recognize that the series is a geometric series:
\[
5 + 5(3) + 5(3)^2 + \ldots
\]
where the first term is \( a = 5 \) and the common ratio is \( r = 3 \).
2. For an infinite geometric series to have a sum, the condition \( |r| < 1 \) must be satisfied. Here:
\[
|r| = |3| = 3 > 1.
\]
3. Since the condition is not met, the series diverges and does not have a finite sum.
\[
\text{No sum}
\]
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Simplify this solution Mind Expander
To compute the sum \( S = 5 + 5(3) + 5(3)^{2} + \ldots \), we can first factor out the common factor of \( 5 \): \[ S = 5(1 + 3 + 3^{2} + \ldots) \] Now, we identify the series inside the parentheses. This is a geometric series with the first term \( a = 1 \) and common ratio \( r = 3 \). A geometric series converges if \( |r| < 1 \). In this case, \( |3| = 3 \) which is greater than 1, so the series diverges. Thus, the original sum does not exist: \[ \text{No sum} \]
