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education/math/MATH1220 (calc II)/Sequences.md

@@ -26,4 +26,16 @@ and say that $a_n$ *converges* to $L$. If no $L$ exists, we say $\{a_n\}$ *diver
 # Sum of an infinite sequence
 - If $f(x)$ is a function and $\{a_n\}$ is a sequence such that $f(n) = a(n)$, then we say $f(x)$ *agrees* with the sequence  $\{a_n\}$ 
 - If $f(x)$ agrees with $\{a_n\}$ then if $\lim_{x \to \infty}f(x) = L$  then $\lim_{n \to \infty}a_n = L$
-- Given the above knowledge, we can apply L'Hospital's rule to sequences of the form $
+- Given the above knowledge, we can apply L'Hospital's rule to sequences that seem to approach $\frac{\infty}{\infty}$.
+
+Remember, L'Hospital's rule states that:
+
+>If you have a limit of the indeterminate form $\dfrac{0}{0}$, the limit can be found by taking the derivative of the numerator, divided by the derivative of the denominator.
+> $$ \lim_{x \to 2} \dfrac{x-2}{x^2-4} = \lim_{x \to 2} \dfrac{1}{2x}$$
+> L'Hospital's Rule can also be used when both the numerator and denominator approach some form of infinity.
+>$$ \lim_{x \to \infty} \dfrac{x^2-2}{3x^2-4} = \lim_{x \to \infty} \dfrac{2x}{6x}$$
+>The above problem can be solved more easily *without* L'Hospital's rule, the leading coefficients are 1/3, so the limit as $x$ approaches $\infty$ is 1/3.
+> L'Hospital's rule **cannot** be used in any other circumstance. 
+
+# Series
+Vocabulary: A **series** is another name for a sum of numbers.