Series tests
by jvadair
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Geometric series converge when
0<\left|r\right|<1\ \ for\ \sum_{n=0}^{\infty}ar^n
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11 cards
| 1 | Geometric series converge when |
0<\left|r\right|<1\ \ for\ \sum_{n=0}^{\infty}ar^n |
| 2 | Sum of a geometric series |
\frac{a}{1-r} |
| 3 | A series diverges by nth term test when |
\lim n\rightarrow\infty\left(a_n\ne0\right)\ for\ \sum_{n=1}^{\infty}a_n |
| 4 | A series converges by the integral test when |
\int_0^{\infty}f\left(x\right)dx\ \ is\ positive\ \&\ finite |
| 5 | The integral test only works when |
f(x) is positive, continuous, and decreasing |
| 6 | A series converges by the p-series test when |
p\ >\ 1\ for\ \sum_{n=1}^{\infty}\frac{1}{n^p} |
| 7 | Direct comparison test |
If a larger function (0 < f1 <= f2) converges/diverges, then the smaller one does too. |
| 8 | The direct/limit comparison tests only work if |
The series has all positive terms |
| 9 | Limit comparison test |
If\lim n\rightarrow\infty\left(\frac{a_n}{b_n}\right)>0,\ the\ a_n\ series\ does\ the\ same\ as\ the\ b_n |
| 10 | Alternating series test |
Converges\ if\ \lim n\rightarrow\infty\left(a_n\right)=0\ \&\ \left|a_{n+1}\right|<\left|a_n\right| |
| 11 | Ratio test |
Converges\ if\ \lim n\rightarrow\infty\left|\frac{a_{n+1}}{a_n}\right|<1,\ diverges\ if\ it's\ >\ 1 |