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education/math/MATH1220 (calc II)/Sequences.md
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@@ -143,3 +143,8 @@ The behavior a given power series falls into one of three cases:
1. Solving for any power series usually starts with the ratio test. 1. Solving for any power series usually starts with the ratio test.
1. Create a ratio with $n + 1$ and $n$. 1. Create a ratio with $n + 1$ and $n$.
$$ \frac{x^{n+1}}{3^{n+1}} \cdot \frac{3^n}{x^n} $$ $$ \frac{x^{n+1}}{3^{n+1}} \cdot \frac{3^n}{x^n} $$
2. Cancel stuff out
$$ \frac{x}{3}$$
2. For a geometric series, it converges when the ratio $r$ is less than one. Written as an equality, this gives us:
$$ |r| < 1 \to |\frac{x}{3}| < 1 \to |x| < 3 $$
This means that the interval of convergence is $(-3, 3)$