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#2: Post edited
by
Snoopy
·
2024-08-11T16:25:12Z (3 months ago)
- **Problem**: Does the series $\sum_{n=1}^{\infty} \frac{n^n}{n!} e^{-n}$ converge?
- **Note**: In many calculus textbook examples, series involving the factorial term $n!$ are typically analyzed using the [ratio test](https://en.wikipedia.org/wiki/Ratio_test). These exercises often skip a detailed examination of how rapidly $n!$ grows. However, the ratio test can often be inconclusive, requiring an understanding of the growth rate of $n!$.
- The given series is a case where the ratio test is inconclusive, and there is no straightforward series available for direct comparison.
I will write my answer below. Different viewpoints or approaches to this problem are welcome.
- **Problem**: Does the series $\sum_{n=1}^{\infty} \frac{n^n}{n!} e^{-n}$ converge?
- **Note**: In many calculus textbook examples, series involving the factorial term $n!$ are typically analyzed using the [ratio test](https://en.wikipedia.org/wiki/Ratio_test). These exercises often skip a detailed examination of how rapidly $n!$ grows. However, the ratio test can often be inconclusive, requiring an understanding of the growth rate of $n!$.
- The given series is a case where the ratio test is inconclusive, and there is no straightforward series available for direct comparison.
- I will write my answers below. Different viewpoints or approaches to this problem are welcome.
#1: Initial revision
by
Snoopy
·
2024-08-11T13:24:14Z (3 months ago)
Does the series $\sum_{n=1}^{\infty} \frac{n^n}{n!} e^{-n}$ converge?
**Problem**: Does the series $\sum_{n=1}^{\infty} \frac{n^n}{n!} e^{-n}$ converge?
**Note**: In many calculus textbook examples, series involving the factorial term $n!$ are typically analyzed using the [ratio test](https://en.wikipedia.org/wiki/Ratio_test). These exercises often skip a detailed examination of how rapidly $n!$ grows. However, the ratio test can often be inconclusive, requiring an understanding of the growth rate of $n!$.
The given series is a case where the ratio test is inconclusive, and there is no straightforward series available for direct comparison.
I will write my answer below. Different viewpoints or approaches to this problem are welcome.