| \(n^{th}\) term test |
\[\sum_{n=1}^{\infty} a_n\] |
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\[\lim\limits_{n \to \infty} a_n \ne 0\] |
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| Geometric |
\[ \sum_{n=1}^{\infty} ar^{n-1}\] \[ \sum_{n=0}^{\infty} ar^{n} \] |
\[ \left|r \right| \lt 1 \] |
\[ \left|r \right| \gt 1 \] |
\[ S_{\infty} = \frac{a}{1-r} \] a is 1st term
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| Telescoping |
\[ \sum_{n=1}^{\infty} \left ( b_n-b_{n+1} \right ) \] |
\[ \lim_{n \to \infty} b_n = L \] |
\[ \] |
\[ S = b_1 - L \] |
| Integral Test |
\[ \sum_{n=1}^{\infty} a_n\] |
\[ \int_{1}^{\infty} f(x)dx \] converges |
\[ \int_{1}^{\infty} f(x)dx \] diverges |
\( a_n \) must be positive, continuous, and decreasing for x \( \geq 1 \) |
| P-series |
\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \] |
\[ p \gt 1 \] |
\[ 0 \lt p \leq 1 \] |
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| Direct Comparison Test |
\[ \sum_{n=1}^{\infty} a_n\] |
\[ 0 \leq a_n \leq b_n \] \( b_n \) converges |
\[ 0 \leq b_n \leq a_n \] \( b_n \) diverges |
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| Limit Comparison Test |
\[ \sum_{n=1}^{\infty} a_n\] |
\[ \lim_{n \to \infty} \frac{a_n}{b_n} = L \] \( b_n \) converges |
\[ \lim_{n \to \infty} \frac{a_n}{b_n} = L \] \( b_n \) diverges |
\[ a_n \gt 0 \] \[ b_n \gt 0 \] \( L \gt 0 \) and finite |
| Alternating Series Test |
\[ \sum_{n=1}^{\infty} (-1)^{n} a_n\] \[ \sum_{n=1}^{\infty} (-1)^{n+1} a_n\] |
\[1. \; \lim_{n \to \infty} a_n = 0 \] \[2. \; a_{n+1} \leq a_{n} \; \forall \; n \geq 1 \]
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| Ratio Test |
\[ \sum_{n=1}^{\infty} a_n\] |
\[ \lim_{n \to \infty} \left | \frac{a_{n+1}}{a_{n}} \right | \lt 1 \] |
\[ \lim_{n \to \infty} \left | \frac{a_{n+1}}{a_{n}} \right | \gt 1 \] |
\[ \lim_{n \to \infty} \left | \frac{a_{n+1}}{a_{n}} \right | = 1 \] test fails |