diff --git a/education/math/MATH1220 (calc II)/Sequences.md b/education/math/MATH1220 (calc II)/Sequences.md index a2d6b1e..91c53ed 100644 --- a/education/math/MATH1220 (calc II)/Sequences.md +++ b/education/math/MATH1220 (calc II)/Sequences.md @@ -142,4 +142,9 @@ The behavior a given power series falls into one of three cases: > When does the series $\sum_{n=0}^\infty \frac{x^n}{3^n}$ converge? 1. Solving for any power series usually starts with the ratio test. 1. Create a ratio with $n + 1$ and $n$. - $$ \frac{x^{n+1}}{3^{n+1}} \cdot \frac{3^n}{x^n} $$ \ No newline at end of file + $$ \frac{x^{n+1}}{3^{n+1}} \cdot \frac{3^n}{x^n} $$ + 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 $(-3, 3)$ \ No newline at end of file