Series (1+1/n)^n, test for divergence

TL;DR

The video discusses using the test for divergence on a series with an exponent of N, concluding that the series diverges.

Transcript

okay what if we have the series as n goes from one to Infinity with 1+ one over n inside and then this time this is raised to the N power only earlier I showed you guys with another Series where we have this similar series but it was raised to the UN Square power right and now series diverges and we use it the root test for it right if you haven't ... Read More

Key Insights

• 🫚 The root test is not suitable for series with an exponent of N.
• 🏆 The test for divergence can be used to determine the convergence or divergence of a series.
• ⛔ The limit of the original series, (1 + 1/n)^n, approaches the value of 'e'.
• 🛀 The test for divergence requires showing that the limit of the series is not equal to zero.
• 🍹 The series with the sum (1 + 1/n)^n diverges based on the application of the test for divergence.
• 🏆 The test for divergence is not applicable to series with an exponent of n^2.
• 🏆 The test for divergence provides a straightforward approach to determining the convergence or divergence of a series.

Questions & Answers

Q: Can we use the root test on a series with an exponent of N?

No, the root test is not applicable to a series with an exponent of N. The limit of the series does not allow us to draw any conclusions.

Q: What other test can we use for this series?

The test for divergence can be used on the series with an exponent of N. By taking the limit of the original series, we find that it approaches the value of 'e'.

Q: Why is it important to show that the limit is not equal to zero when using the test for divergence?

When using the test for divergence, it is crucial to demonstrate that the limit of the series is not equal to zero. This is because a zero limit would not allow us to draw any conclusions about the convergence or divergence of the series.

Q: What is the conclusion drawn from applying the test for divergence on this series?

The conclusion is that the series, with the sum from 1 to infinity of (1 + 1/n)^n, diverges. This is determined by showing that the limit of the series is not equal to zero.

Summary & Key Takeaways

• The video explores the possibility of using the root test on a series with an exponent of N, but concludes that it is not applicable.

• The test for divergence is then used on the series, and it is found that the limit of the series does not equal zero.

• Based on the test for divergence, it is concluded that the series diverges.