Integral Test, Convergence, Infinite Series - Calculus

The integral test determines if an infinite series converges or diverges by comparing it to an improper integral. The test applies if the series' corresponding function, \(f(x)\), is positive, continuous, and decreasing on the interval \([1,\infty )\). If the improper integral \(\int _{1}^{\infty }f(x)\,dx\) converges to a finite value, then the series also converges. If the integral diverges, then the series diverges as well.

💡How to use the integral test
• Formulate the function: Convert the series \(\sum _{n=1}^{\infty }a_{n}\) into a function \(f(x)\) by replacing \(n\) with \(x\).
• Check the conditions: Verify that the function \(f(x)\) is positive, continuous, and decreasing for all \(x\ge 1\). If the function is not decreasing from the start, it can be tested for "eventual" decreasing and positive behavior on some interval \([N,\infty )\), as the convergence of the integral and series will not be affected by a finite number of terms.
• Evaluate the improper integral: Compute the improper integral \(\int _{1}^{\infty }f(x)\,dx\). This is done by evaluating the limit: \(\lim _{R\rightarrow \infty }\int _{1}^{R}f(x)\,dx\).
• Determine convergence:
◦ If the integral equals a finite number (it converges), then the original series also converges.
◦ If the integral does not equal a finite number (it diverges, i.e., goes to \(\infty \) or \(-\infty \)), then the original series also diverges.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1DlYi7xcmyEAhWTnwUN8zRgisBdHIdmtr/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1VEHFVDs67vvvQVVsozjmOCO5RbEgzbkA/view?usp=drive_link

💡Chapters:
00:00 Integral test for convergence
01:38 Worked examples

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