Sequences and Series

Convergence and Divergence

Understanding the behavior of infinite sequences and series

Monotonic Sequences

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 Sequences

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

Series Tests

Tools for determining convergence of infinite series

Ratio Test

\[\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = L\]

L < 1: Converges

L > 1: Diverges

L = 1: Inconclusive

Root Test

\[\lim_{n \to \infty} \sqrt[n]{|a_n|} = L\]

L < 1: Converges

L > 1: Diverges

L = 1: Inconclusive

Integral Test

For continuous, decreasing, positive \(f(x)\):

\[\sum_{n=1}^{\infty} f(n) \text{ converges } \iff \int_1^{\infty} f(x)\,dx \text{ converges}\]

Power Series

Understanding radius of convergence and series representations

Radius of Convergence

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: \[R = \frac{1}{\lim_{n \to \infty} \sqrt[n]{|c_n|}}\]

Taylor and Maclaurin Series

Representing functions as infinite series

Taylor Series Formula

\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\]

Common Maclaurin 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:

\[\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots, |x| < 1\]