| Test Name | Description | Comment |
|---|---|---|
| Test for Divergence for all series (p. 566) | \(\sum\limits_{k=1}^{\infty }\,a_{k}\) diverges if \(\lim\limits_{n\,\rightarrow \,\infty}a_{n}\neq 0\). | No information is obtained about convergence if \(\lim\limits_{n\,\rightarrow \,\infty}a_{n}=0\). |
| Integral Test (p. 569) for series of positive terms | \(\sum\limits_{k=1}^{\infty }\,a_{k}\) converges (diverges) if \(\int_{1}^{\infty}\,f(x)\,dx\) converges (diverges), where \(f\) is continuous, positive, and decreasing for \(x\geq 1;\) and \(f(k)=a_{k}\) for all \(k\). | Good to use if \(f\) is easy to integrate. |
| Comparison Test for Convergence for series of positive terms (p. 576) | \(\sum\limits_{k=1}^{\infty }\,a_{k}\) converges if \(0< a_{k}\leq b_{k}\) and the series \(\sum\limits_{k=1}^{\infty }b_{k}\) converges. | \(\sum\limits_{k=1}^{\infty }b_{k}\) must have positive terms and be convergent. |
| Comparison Test for Divergence for series of positive terms (p. 576) | \(\sum\limits_{k=1}^{\infty}\,a_{k}\) diverges if \(a_{k}\geq c_{k}> 0\) and the series \(\sum\limits_{k=1}^{\infty }c_{k}\) diverges. | \(\sum\limits_{k=1}^{\infty }c_{k}\) must have positive terms and be divergent. |
| Limit Comparison Test (p. 577) for series of positive terms | \(\sum\limits_{k=1}^{\infty}\,a_{k}\) converges (diverges) if \(\sum\limits_{k=1}^{\infty }b_{k}\) converges (diverges), and \(\lim\limits_{n\,\rightarrow \,\infty }\dfrac{a_{n}}{b_{n}}=L,\) a positive real number. | \(\sum\limits_{k=1}^{\infty}b_{k}\) must have positive terms, whose convergence (divergence) can be determined. |
| Alternating Series Test (p. 582) | \(\sum\limits_{k=1}^{\infty}(-1)^{k+1}a_{k}\), \(a_{k}>0,\) converges if
| The error made by using the \(n\)th partial sum as an approximation to the sum \(S\) of the series is less than the \((n+1) \)st term of the series. |
| Absolute Convergence Test (p. 586) | If \(\sum\limits_{k=1}^{\infty}\,\left\vert a_{k}\right\vert \) converges, then \(\sum\limits_{k=1}^{\infty}a_{k}\) converges. | The converse is not true. That is, if \(\sum\limits_{k=1}^{\infty}\,\left\vert a_{k}\right\vert \) diverges, \(\sum\limits_{k=1}^{\infty }a_{k}\) may converge. |
| Ratio Test (p. 591) for series with nonzero terms | \(\sum\limits_{k=1}^{\infty }a_{k}\) converges if \(\lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert <1\). \(\sum\limits_{k=1}^{\infty}a_{k}\) diverges if \(\lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert >1\) or if \(\lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert = \infty\). | Good to use if \(a_{n}\) includes factorials or powers. It provides no information if \(\lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert =1.\) |
| Root Test (p. 593) for series with nonzero terms | \(\sum\limits_{k=1}^{\infty }a_{k}\) converges if \(\lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\vert a_{n}\vert }<1\). \( \sum\limits_{k=1}^{\infty }a_{k}\) diverges if \(\lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\vert a_{n}\vert }>1\) or if \(\lim\limits_{n\,\rightarrow \,\infty }\sqrt[n]{\vert a_{n}\vert }=\infty\). | Good to use if \(a_{n}\) involves \(n\)th powers. It provides no information if \(\lim\limits_{n\,\rightarrow\,\infty }\sqrt[n]{\vert a_{n}\vert }=1\). |