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0$ or does not exist, then divergent \\\hline Integral & $f(n)=a_{n}$ is continuous, positive, decreasing function $\forall x\geq 1$ and $\int_{1}^{\infty}f(x)dx$ converges else divergent \\\hline p-series & $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$convergent $\leftrightarrow p > 1$ \\\hline - Ratio & $0 \geq \lim\limits_{n \to \infty} |\frac{a_{n+1}}{a_{n}}|=L < 1$ abs. convergent, $> 1$ divergent, $= 1$ inconclusive \\\hline - Root & $0 \geq \lim\limits_{n \to \infty} \sqrt[n]{a_{n}}=L < 1$ abs. convergent, $> 1$ divergent, $= 1$ inconclusive \\\hline + Harmonic Series & $\sum_{n=1}^{\infty} \frac{1}{n}$ divergent \\\hline + Ratio \tiny{If Factorial} & $0 \geq \lim\limits_{n \to \infty} |\frac{a_{n+1}}{a_{n}}|=L < 1$ abs. convergent, $> 1$ divergent, $= 1$ inconclusive \\\hline + Root \tiny{If nth power} & $0 \geq \lim\limits_{n \to \infty} \sqrt[n]{a_{n}}=L < 1$ abs. convergent, $> 1$ divergent, $= 1$ inconclusive \\\hline Alternating series & $b_{n}$ decreasing, $\lim\limits_{n \to \infty}b_{n} = 0$, then $\sum_{n=1}^{\infty}(-1)^{n-1}b_{n} = b_{1}-b_{2}+b_{3}... $ is convergent \\\hline Power Series & $b_{n}$ decreasing, $\lim\limits_{n \to \infty}b_{n} = 0$, then $\sum_{n=1}^{\infty}(-1)^{n-1}b_{n} = b_{1}-b_{2}+b_{3}... $ is convergent \\\hline + Comparison Test & Compare to well known series such as p-series, harmonic \\\hline \end{tabular} \end{center} @@ -445,6 +447,5 @@ For $-1 < x < 1$ \\ & $= 1 + kx + \frac{k(k-1)}{2!}x^2 + \dots$ \end{tabular} - \end{multicols*} \end{document}