📘 Calculus 2: Infinite Series Study Notes

Geometric Series Sum formula

$$\frac{a}{1 - r}$$

P-Series Convergence condition

$$p > 1$$

Harmonic Series formula

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

Direct Comparison (proving convergence)

If $0 \leq a<em>n \leq b</em>n$ and $\sum b_n$ converges.

Sequence

An ordered list of numbers; convergence means $\lim<em>{n \to \infty} a</em>n = L$.

Series

The sum of the terms represented by $\sum a_n$.

P-Series General Form

$\sum_{n=1}^{\infty} \frac{1}{n^p}$.

nth Term Test (Divergence Test)

If $\lim<em>{n \to \infty} a</em>n \neq 0$, then $\sum a_n$ diverges.

The Integral Test

$\sum a<em>n$ and $\int</em>{1}^{\infty} f(x) dx$ behave the same way under certain conditions.

Requirements for the Integral Test

Function $f(x)$ must be Positive, Continuous, and Decreasing for $x \geq 1$.

Telescoping Series

Use Partial Fraction Decomposition to cancel middle terms, leaving a finite sum $S_n$.

Geometric Series Convergence condition

Converges if $|r| < 1$.

Harmonic Series Divergence

The Harmonic Series diverges because $p = 1$.