Understanding the behavior of infinite sequences and series
A sequence that consistently increases or decreases:
Increasing: \(a_{n+1} > a_n\)
Decreasing: \(a_{n+1} < a_n\)
Example: \(\left\{\frac{1}{n}\right\}\) decreases and converges to 0
Bounded Above: \(a_n \leq M\) for some \(M\)
Bounded Below: \(a_n \geq m\) for some \(m\)
Example: \(\{(-1)^n\}\) is bounded but oscillates
Tools for determining convergence of infinite series
L < 1: Converges
L > 1: Diverges
L = 1: Inconclusive
L < 1: Converges
L > 1: Diverges
L = 1: Inconclusive
For continuous, decreasing, positive \(f(x)\):
\[\sum_{n=1}^{\infty} f(n) \text{ converges } \iff \int_1^{\infty} f(x)\,dx \text{ converges}\]Understanding radius of convergence and series representations
For series \(\sum_{n=0}^{\infty} c_n(x-a)^n\):
Using Ratio Test: \[R = \frac{1}{\lim_{n \to \infty} \left|\frac{c_{n+1}}{c_n}\right|}\]
Using Root Test: \[R = \frac{1}{\lim_{n \to \infty} \sqrt[n]{|c_n|}}\]
Representing functions as infinite series
Exponential:
\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\]Sine:
\[\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\]Cosine:
\[\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\]Natural Log:
\[\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots, |x| < 1\]Test your understanding of sequences and series concepts