vault backup: 2025-09-26 12:32:34

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

@@ -22,3 +22,8 @@ and say that $a_n$ *converges* to $L$. If no $L$ exists, we say $\{a_n\}$ *diver
 5. If $L = M$ and a sequence $c_n$ exists such that $a_n \le c_n \le b_n$ for all $n$, then $c_n \to L = M$
 6. If $a_n$ and $b_n$ both approach infinity at a similar rate, $\frac{a_n}{b_n}$  will approach an arbitrary value. This value can be found by rewriting $\frac{a_n}{b_n}$ in such a manner that the end behavior of the series is more easily identifiable
 	> For example, given the series $c_n = \frac{n}{2n+1}$, both the numerator and the denominator approach infinity at a similar rate. However, when the numerator and denominator are both multiplied by $\frac{1}{n}$, it becomes $\frac{1}{2+\frac{1}{n}}$, an equivalent sequence that more clearly converges on $1/2$. 
+
+# 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 $