Does the Series SUM(e^n/3^(n1)) Converge or Diverge? If it Converges, Find the Sum.  Summary and Q&A
TL;DR
Determine whether the infinite geometric series converges or diverges and find the sum if it converges.
Key Insights
 🥳 Infinite geometric series can be identified by their common ratio and first term.
 💁 Manipulation can be done to rewrite a series in a geometric form.
 🥳 The geometric series test helps determine if an infinite series converges or diverges based on the absolute value of the common ratio.
Transcript
in this problem we have an infinite geometric series and the question wants us to determine if it converges or diverges and if it converges we're being asked to find the sum of the series so this appears to be geometric so geometric series generally look like this something like a r to the n or something like this a r to the n minus ones any of the... Read More
Questions & Answers
Q: How is an infinite geometric series represented?
An infinite geometric series can be represented as a sum from 1 to infinity of a times r^(n1), where a is the first term and r is the common ratio.
Q: What manipulation is done to rewrite the series in a geometric form?
To rewrite the series as a geometric series, the numerator and denominator are separated, and the denominator is simplified by multiplying the powers of 3.
Q: What is the geometric series test?
The geometric series test states that if the absolute value of the common ratio is less than 1, the series converges. If the absolute value is greater than or equal to 1, the series diverges.
Q: How is the sum of the infinite geometric series found?
The sum is found by plugging the value of the common ratio into the formula, which is the first term multiplied by the common ratio divided by (1  common ratio).
Summary & Key Takeaways

The problem involves determining if an infinite geometric series converges or diverges and finding the sum of the series.

Manipulation is done to rewrite the series in a geometric form, and the geometric series test is used to determine convergence.

The absolute value of the common ratio is compared to 1 to determine whether the series converges or diverges.

If the series converges, the sum can be found by substituting the common ratio into the formula.