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17Calculus Infinite Series Table

Single Variable Calculus
Multi-Variable Calculus

Here is a table of series and tests. The table is arranged in order of use, as described on the page outlining how to choose a test. If you are already familiar with the tests, there is enough detail in this table to allow you to use it rather than flipping through the pages in your textbook. You may download a pdf version of this table. Make as many copies as you want and share it with your friends and classmates. For teachers, feel free to make copies for your students and use it in your class. All we ask (from both students and teachers) is that you keep the 17calculus.com information at the bottom of the page.

Infinite Series Table

Group 1 - First Test You Should Always Try

Divergence (nth-Term) Test
\(\displaystyle{ \sum_{n=1}^{\infty}{a_n} }\)

convergence: cannot be used

divergence:   \(\displaystyle{ \lim_{n \to \infty}{ a_n \neq 0} }\)

Group 2 - Special Series

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^p} } }\)

convergence: \( p > 1 \)

divergence:   \( p \leq 1 \)

Geometric Series
\(\displaystyle{ \sum_{n=0}^{\infty}{ ar^n } }\)

convergence: \( |r| < 1 \)

divergence:   \( |r| \geq 1 \)

sum: \(\displaystyle{ S = \frac{a}{1-r}; ~~ |r| > 0 }\)

Alternating Series
\(\displaystyle{ \sum_{n=1}^{\infty}{ (-1)^n a_n } }\)

convergence: 1. \( \lim_{n \to \infty}{a_n=0}\) and 2. \(0 < a_{n+1} \leq a_n \)

divergence:   cannot be used

condition 1 is the divergence test; remainder: \( |R_N| \leq a_{N+1} \)

Telescoping Series
\(\displaystyle{ \sum_{n=1}^{\infty}{ (b_n - b_{n+1}) } }\)

convergence: \(\displaystyle{ \lim_{n \to \infty}{b_n} = L }\)

divergence:   cannot be used

\(L\) is finite; sum: \(S=b_1 - L\)

Group 3 - Core Series Tests (in order of use)

Ratio Test
\(\displaystyle{ \sum_{n=1}^{\infty}{a_n} }\)

convergence: \(\displaystyle{ \lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right| } < 1 }\)

divergence:   \(\displaystyle{ \lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right| } > 1 }\)

test is inconclusive if \(\displaystyle{ \lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right| } = 1 }\)

Limit Comparison Test
\(\displaystyle{ \sum_{n=1}^{\infty}{a_n} }\)

convergence: \(\displaystyle{ 0 \leq \lim_{n \to \infty}{ \frac{a_n}{t_n} } < \infty }\) and \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) converges

divergence:   \(\displaystyle{ \lim_{n \to \infty}{ \frac{a_n}{t_n} } > 0 }\) and \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) diverges

\(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) is the test series, \(a_n > 0\) and \(t_n > 0\)

Direct Comparison Test
\(\displaystyle{ \sum_{n=1}^{\infty}{a_n} }\)

convergence: \( 0 < a_n \leq t_n \) and \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) converges

divergence:   \( 0 < t_n \leq a_n \) and \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\) diverges

test series: \(\displaystyle{ \sum_{n=1}^{\infty}{t_n} }\); \(a_n, t_n > 0\)

Integral Test
\(\displaystyle{ \sum_{n=1}^{\infty}{a_n} }\)

convergence: \(\displaystyle{ \int_{k}^{\infty}{f(x)~dx} }\) converges

divergence:   \(\displaystyle{ \int_{k}^{\infty}{f(x)~dx} }\) diverges

\(f(x)\) is continuous, positive and decreasing for \(x > k\); \(a_n = f(n) \geq 0 \)

Root Test
\(\displaystyle{ \sum_{n=1}^{\infty}{a_n} }\)

convergence: \(\displaystyle{ \lim_{n \to \infty}{ \sqrt[n]{|a_n|} } < 1 }\)

divergence:   \(\displaystyle{ \lim_{n \to \infty}{ \sqrt[n]{|a_n|} } > 1 }\)

inconclusive: \(\displaystyle{ \lim_{n \to \infty}{ \sqrt[n]{|a_n|} } = 1 }\)

Really UNDERSTAND Calculus

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