What series convergence test do I use?

The speaker begins a live stream session, introducing the topic of series convergence tests. They mention that they will be going over 15 questions related to series tests, with a downloadable file provided in the description for reference. The session is aimed at students who might be preparing for a test, and the speaker hints at the possibility of some questions being similar to those on an upcoming test. They also provide a 'secret weapon' in the form of notes on various series convergence tests, available for download, and engage with the audience, mentioning the time and location of some viewers.

The speaker presents the first series convergence question involving the cosine function and asks the audience to determine if it converges or diverges. After a brief interaction, the correct answer is provided by a participant named Emilio. The speaker then explains that the series diverges by using the test for divergence, emphasizing the importance of showing that the limit of the series' term does not equal zero.

The speaker moves on to the second question, which involves a geometric series. They guide the audience through simplifying the series and applying the geometric series test. The correct answer is that the series converges, and the speaker provides the value to which it converges, emphasizing the importance of showing that the absolute value of the common ratio is less than one.

The third question is tackled using the ratio test. The series involves a factorial term, which is a good indicator for using the ratio test. The speaker demonstrates the algebraic steps to simplify the series and calculate the limit needed for the ratio test. The conclusion is that the series converges, and the speaker highlights the importance of the factorial term in determining the appropriate test.

The fourth question leads to an indeterminate form when using the ratio test. The speaker discusses the mistake of prematurely concluding the limit and instead introduces a 'secret weapon' fact about limits involving one to the power of infinity. They apply this fact to find that the series converges by the ratio test, providing an alternative method when direct computation is not straightforward.

The speaker addresses the fifth question, which involves sine in the series. They suggest using the absolute value to simplify the series and check for absolute convergence. By comparing the series to a simpler convergent series, the speaker demonstrates the use of the direct comparison test to conclude that the original series also converges.

In the sixth question, the speaker cautions against the incorrect application of the limit comparison test. They illustrate a common mistake and then provide a hint for the correct approach, emphasizing the importance of using the appropriate series for comparison in the test.

The speaker corrects the approach and applies the limit comparison test properly to the sixth question. They identify the dominating terms in the series and simplify the expression before taking the limit. The conclusion is that the series converges, and the speaker reminds the audience to state that the limit is greater than zero when using the limit comparison test.

The speaker introduces the integral test for the seventh question, which involves a natural logarithm term. They outline the conditions for the integral test and proceed to calculate the improper integral. The result is that the integral diverges, leading to the conclusion that the series also diverges.

For the eighth question, the speaker uses the test for divergence on a series with an exponentiated term. They take the limit of the series' term as n approaches infinity and show that it does not equal zero, leading to the conclusion that the series diverges.

The ninth question involves a series with a sine squared term. The speaker uses the limit comparison test, comparing the series to a known divergent series of 1/n^2. They demonstrate that the limit of the comparison yields a value greater than zero, indicating that the original series also converges.

The speaker tackles the tenth question, which includes a series with factorials in the terms. They apply the ratio test, carefully breaking down the factorial expressions and calculating the limit. The result is a limit greater than one, indicating that the series diverges.

In the eleventh question, the speaker identifies a series that can be expressed in sigma notation as a geometric series. They find the common ratio and show that its absolute value is less than one, leading to the conclusion that the series converges.

The twelfth question involves a series with inverse tangent functions. The speaker uses the integral test, setting up the integral of the function and calculating it. The integral converges, indicating that the series also converges.

For the thirteenth question, the speaker applies the alternating series test to a series with an alternating pattern. They check the limit of the series' terms and confirm that they are decreasing, leading to the conclusion that the series converges.

The fourteenth question is a telescoping series, which the speaker identifies and simplifies using partial fractions. They demonstrate the cancellation of terms and calculate the sum of the series, showing that it converges to a specific value.

In the fifteenth and final question, the speaker presents a series with terms involving the natural logarithm. They apply the p-series test, identifying the value of p and concluding that the series diverges because p is greater than one.

As a bonus, the speaker poses a question involving a series with cosine terms. They hint at using a trigonometric identity to compare the series to one that was discussed earlier in the session, suggesting that the audience should be able to determine the convergence or divergence of the series based on previous examples.

The speaker concludes the session by summarizing the topics covered and expressing hope that the session was helpful for the audience. They also hint at the possibility of future sessions, inviting the audience to stay tuned for more on series convergence tests.