★ Must-know
For r>0, the sequence 1/n^r approaches 0 as n approaches infinity.
The sequence r^n converges precisely when -1<r<=1; its limit is 0 for -1<r<1 and 1 for r=1.
Further detail
📌 If a_n approaches L and f is continuous at L, then f(a_n) approaches f(L).
1. What mathematical object is an infinite sequence?
2. For which values of r does the sequence r^n converge?
3. Which condition guarantees that a sequence converges?
What is an infinite sequence in mathematics?
An infinite sequence is an ordered list of numbers indexed by integers.
When does a sequence a_n converge to a limit L?
When its terms become arbitrarily close to L as n becomes large.
What defines a monotonic sequence?
It is either increasing with a_n < a_(n+1) or decreasing with a_n > a_(n+1).
What does every bounded monotonic sequence do?
It converges.
Sequence limit
Sequence terms approach L as n→∞.
Monotone + bounded
Always convergent sequences.
It covers the core concepts of Infinite Series and Taylor Methods, organized by topic with the key definitions, explanations and takeaways, so it's easier to learn and remember.
Read the full study guide →The practice quiz has 10 multiple-choice questions, each with an explanation. Use it to check what you know and find your gaps.
Take the quiz (10 questions) →There are 11 flashcards on Infinite Series and Taylor Methods, with a question on the front and the answer on the back. Try to recall each answer before you flip, and let spaced repetition bring back the ones you miss.
See all 11 flashcards →Upload a PDF or paste your notes. Revizly writes a study guide, a practice quiz and flashcards in about a minute.