Infinite Sequences
Infinite sequences, a fundamental concept in mathematics, hold a captivating allure for their unbounded nature. These sequences, extending indefinitely in one direction, present a rich landscape of patterns, convergence, and divergent behaviors. From the simple arithmetic progression to the intricate Fibonacci sequence and beyond, infinite sequences permeate various branches of mathematics, physics, and computer science, offering insights into the structure of numbers, functions, and algorithms. Exploring their properties unlocks a realm of exploration where infinite possibilities unfold, challenging our understanding and igniting curiosity about the infinite expanse of mathematical phenomena.
Questions
- Does #a_n=(2+n+(n^3))/sqrt(2+(n^2)+(n^8)) #converge? If so what is the limit?
- Does #a_n=n*{(3/n)^(1/n)} #converge? If so what is the limit?
- What is the formula for the sequence #2, -1, 4, -7, 10, -13, 16,...# ?
- How do you determine whether the sequence #a_n=(-1)^n/sqrtn# converges, if so how do you find the limit?
- Does #a_n=(n + (n/2))^(1/n) # converge?
- How do you determine whether the sequence #a_n=n!-10^n# converges, if so how do you find the limit?
- How do you determine if #a_n=1-1.1+1.11-1.111+1.1111-...# converge and find the sums when they exist?
- How do you find the first three iterate of the function #f(x)=3x+5# for the given initial value #x_0=-4#?
- If #a_n# converges and #lim_(n->oo) a_n -b_n=c#, where c is a constant, does #b_n# converge?
- What is the pattern in the sequence 100, 19, 83, 34, 70, 45?
- What is an alternating sequence?
- How do you determine whether the infinite sequence #a_n=n+1/n# converges or diverges?
- How do you determine if #a_n=6+19+3+4/25+8/125+16/625+...+(2/5)^n+...# converge and find the sums when they exist?
- How do you find the first three iterate of the function #f(x)=2x^2-5# for the given initial value #x_0=-1#?
- What does it mean for a sequence to be monotone?
- How do you determine if #a_n=1+3/7+9/49+...+(3/7)^n+...# converge and find the sums when they exist?
- What is the difference between an infinite sequence and an infinite series?
- Does #a_n=x^n/(n!) # converge for any x?
- What is the root test?
- Does #a_n=x^n/n^x # converge for any x?