Example 4: Series with terms (ln n) / n diverges by the Integral Test - Hive - YouTube - Notes - Playlist - Sequences and Series

In this video, I show that the sum of the series with terms (ln n) / n diverges to infinity by computing its corresponding improper integral via the Integral Test. First noting that the natural logarithm function is continuous, and positive for x is greater than 1, then so is the corresponding function f(x) = (ln x) / x. However, it is uncertain whether the function is increasing or decreasing, so I take the derivative and show that it is negative for x is greater than the number e, thus the function itself is also decreasing. This allows us to apply the Integral Test for convergence and the corresponding improper integral approaches infinity, thus our series diverges!

#math #calculus #series #logarithms #education

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Remainder Estimate for the Integral Test + Example - Hive - YouTube - Notes - Playlist - Sequences and Series

In this video, I approximate the accuracy of the partial sums of a series that converges via the Integral Test by estimating the size of the remainder. This is done by comparing the areas of the rectangles below the curve whose function corresponds to the terms of the series, but doing so for all the terms after the n-th partial sum. The areas of the rectangles will be greater than the improper integral starting from n and less than the improper integral starting from n + 1; this is the Remainder Estimate for the Integral Test!

I illustrate this with an example on the series with terms 1/n³ and show that with 32 terms in the partial sum, the accuracy is within 0.0005 of the actual sum of the infinite series.

#math #calculus #series #remainder #education