Infinite Sequence, Definition, Convergence, Practice Problems - Calculus

An infinite sequence converges if its terms approach a specific, finite number (its limit) as the index \(n\) approaches infinity. An infinite sequence diverges if the limit does not exist, or if it increases or decreases without bound. To determine convergence, you find the limit of the sequence's general term as \(n\) approaches infinity; if this limit is a finite number, the sequence converges to that number.

💡Convergent sequences
• Definition: A sequence \(a_{n}\) converges to a limit \(L\) if \(\lim _{n\rightarrow \infty }a_{n}=L\), where \(L\) is a finite number.
• Meaning: As \(n\) gets larger and larger, the terms of the sequence get closer and closer to \(L\).
• Example: The sequence \(a_{n}=1/n\) converges because as \(n\) approaches infinity, \(1/n\) approaches \(0\).

💡Divergent sequences
• Definition: A sequence diverges if it does not have a finite limit.
• Meaning: The terms of the sequence either:
⚬Do not approach a single, finite value.
⚬Increase or decrease without any upper or lower bound (i.e., go to infinity).
• Example: The sequence \(a_{n}=n^{2}\) diverges because as \(n\) approaches infinity, the terms grow infinitely large.

💡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 Infinite sequences
00:57 Convergence
02:15 Worked examples

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