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.
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Group 1  First Test You Should Always Try  

Divergence (nthTerm) Test  
convergence: cannot be used  
divergence: \(\displaystyle{ \lim_{n \to \infty}{ a_n \neq 0} }\)  
Group 2  Special Series  
pSeries  
convergence: \( p > 1 \)  
divergence: \( p \leq 1 \)  
Geometric Series  
convergence: \( r < 1 \)  
divergence: \( r \geq 1 \)  
sum: \(\displaystyle{ S = \frac{a}{1r}; ~~ r > 0 }\)  
Alternating Series  
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  
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  
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  
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  
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  
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  
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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