Calculus II

Arc Length Formula

$L=\int_{a}^{b}\sqrt{1+\left (\frac{dy}{dx} \right )^{2}}dx, a\leq x\leq b$

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L’Hopital’s Rule

$\lim_{x \to c}\frac{f(x)}{g(x)}=\lim_{x \to c}\frac{f'(x)}{g'(x)}$

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Monotonic Sequences

A monotonic sequence is one where its terms are nondecreasing or nonincreasing.

$a_1\leq a_2\leq a_3\leq \cdots \leq a_n\leq \cdots$ (nondecreasing sequence)

$a_1\geq a_2\geq a_3\geq \cdots \geq a_n\geq \cdots$ (nonincreasing sequence)

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Geometric Series

$\sum_{n=0}^{\infty}ar^n=a+ar+ar^2+\cdots +ar^n+\cdots ,a\neq 0$

A geometric series diverges if $\left | r \right |\geq1$ .

If $0< \left | r \right |< 1$ , then the geometric series converges to

$\sum_{n=0}^{\infty }ar^n=\frac{a}{1-r}$

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The nth-Term Test for Divergence

If $\sum_{n=1}^{\infty }a_n$ , then $\lim_{n \to \infty}a_n=0$ .

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The Integral Test

If $\int_{1}^{\infty}f(x)dx$ diverges (or converges), then $\sum_{n=1}^{\infty}a_n$ also diverges (or converges).

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