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Convergence tests
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<h2 id="list-of-tests">List of tests</h2> <h3><a href="/facts/Term_test/9lc5wh7m">Limit of the summand</a></h3> <p>If the limit of the summand is undefined or nonzero, that is lim n → ∞ a n ≠ 0 {\displaystyle \lim _{n\to \infty }a_{n}\neq 0} , then the series must diverge. In this sense, the partial sums are <a href="/facts/Cauchy_sequence/7Q5c8TuF">Cauchy</a> <a href="/facts/Only_if/bYSxGJ66">only if</a> this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test. </p> <h3><a href="/facts/Ratio_test/SpvhbMQF">Ratio test</a></h3> <p>This is also known as d'Alembert's criterion. </p> Consider two limits ℓ = lim inf n → ∞ | a n + 1 a n | {\displaystyle \ell =\liminf _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|} and L = lim sup n → ∞ | a n + 1 a n | {\displaystyle L=\limsup _{n\to \infty }\left|{\frac {a_{n+1}}{a_{n}}}\right|} . If ℓ > 1 {\displaystyle \ell >1} , the series diverges. If L < 1 {\displaystyle L<1} then the series converges absolutely. If ℓ ≤ 1 ≤ L {\displaystyle \ell \leq 1\leq L} then the test is inconclusive, and the series may converge absolutely, conditionally or diverge. <h3><a href="/facts/Root_test/rUJKuqkl">Root test</a></h3> <p>This is also known as the <i>n</i>th root test or Cauchy's criterion. </p> Let r = lim sup n → ∞ | a n | n , {\displaystyle r=\limsup _{n\to \infty }{\sqrt[{n}]{|a_{n}|}},} where lim sup {\displaystyle \limsup } denotes the <a href="/facts/Limit_superior/QeAhlygs">limit superior</a> (possibly ∞ {\displaystyle \infty } ; if the limit exists it is the same value). If <i>r</i> < 1, then the series converges absolutely. If <i>r</i> > 1, then the series diverges. If <i>r</i> = 1, the root test is inconclusive, and the series may converge or diverge. <p>The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h3><a href="/facts/Integral_test/7KRhJwRr">Integral test</a></h3> <p>The series can be compared to an integral to establish convergence or divergence. Let f : [ 1 , ∞ ) → R + {\displaystyle f:[1,\infty )\to \mathbb {R} _{+}} be a non-negative and <a href="/facts/Monotonic_function/MjQK0YL2">monotonically decreasing function</a> such that f ( n ) = a n {\displaystyle f(n)=a_{n}} . If ∫ 1 ∞ f ( x ) d x = lim t → ∞ ∫ 1 t f ( x ) d x < ∞ , {\displaystyle \int _{1}^{\infty }f(x)\,dx=\lim _{t\to \infty }\int _{1}^{t}f(x)\,dx<\infty ,} then the series converges. But if the integral diverges, then the series does so as well. In other words, the series a n {\displaystyle {a_{n}}} converges <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the integral converges. </p> <h4>p-series test</h4> <p>A commonly-used corollary of the integral test is the p-series test. Let k > 0 {\displaystyle k>0} . Then ∑ n = k ∞ ( 1 n p ) {\displaystyle \sum _{n=k}^{\infty }{\bigg (}{\frac {1}{n^{p}}}{\bigg )}} converges if p > 1 {\displaystyle p>1} . </p><p>The case of p = 1 , k = 1 {\displaystyle p=1,k=1} yields the harmonic series, which diverges. The case of p = 2 , k = 1 {\displaystyle p=2,k=1} is the <a href="/facts/Basel_problem/2PManmlC">Basel problem</a> and the series converges to π 2 6 {\displaystyle {\frac {\pi ^{2}}{6}}} . In general, for p > 1 , k = 1 {\displaystyle p>1,k=1} , the series is equal to the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> applied to p {\displaystyle p} , that is ζ ( p ) {\displaystyle \zeta (p)} . </p> <h3><a href="/facts/Direct_comparison_test/AeHYrl7o">Direct comparison test</a></h3> <p>If the series ∑ n = 1 ∞ b n {\displaystyle \sum _{n=1}^{\infty }b_{n}} is an <a href="/facts/Absolutely_convergent/BXz9kBJ5">absolutely convergent</a> series and | a n | ≤ | b n | {\displaystyle |a_{n}|\leq |b_{n}|} for sufficiently large <i>n</i> , then the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges absolutely. </p> <h3><a href="/facts/Limit_comparison_test/NjqDPkmo">Limit comparison test</a></h3> <p>If { a n } , { b n } > 0 {\displaystyle \{a_{n}\},\{b_{n}\}>0} , (that is, each element of the two sequences is positive) and the limit lim n → ∞ a n b n {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}} exists, is finite and non-zero, then either both series converge or both series diverge. </p> <h3><a href="/facts/Cauchy_condensation_test/s7CoyYeY">Cauchy condensation test</a></h3> <p>Let { a n } {\displaystyle \left\{a_{n}\right\}} be a non-negative non-increasing sequence. Then the sum A = ∑ n = 1 ∞ a n {\displaystyle A=\sum _{n=1}^{\infty }a_{n}} converges <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the sum A ∗ = ∑ n = 0 ∞ 2 n a 2 n {\displaystyle A^{*}=\sum _{n=0}^{\infty }2^{n}a_{2^{n}}} converges. Moreover, if they converge, then A ≤ A ∗ ≤ 2 A {\displaystyle A\leq A^{*}\leq 2A} holds. </p> <h3><a href="/facts/Abel%2527s_test/uPBmEV6N">Abel's test</a></h3> <p>Suppose the following statements are true: </p> <ol><li> ∑ a n {\displaystyle \sum a_{n}} is a convergent series,</li> <li> { b n } {\displaystyle \left\{b_{n}\right\}} is a monotonic sequence, and</li> <li> { b n } {\displaystyle \left\{b_{n}\right\}} is bounded.</li></ol> <p>Then ∑ a n b n {\displaystyle \sum a_{n}b_{n}} is also convergent. </p> <h3><a href="/facts/Absolute_convergence/BXz9kBJ5">Absolute convergence test</a></h3> <p>Every <a href="/facts/Absolutely_convergent/BXz9kBJ5">absolutely convergent</a> series converges. </p> <h3><a href="/facts/Alternating_series_test/MHYqbQp4">Alternating series test</a></h3> <p>Suppose the following statements are true: </p> <ul><li> ( a n ) n = 1 ∞ {\displaystyle (a_{n})_{n=1}^{\infty }} is monotonic,</li> <li> lim n → ∞ a n = 0 {\displaystyle \lim _{n\to \infty }a_{n}=0} </li></ul> <p>Then ∑ n = 1 ∞ ( − 1 ) n a n {\displaystyle \sum _{n=1}^{\infty }(-1)^{n}a_{n}} and ∑ n = 1 ∞ ( − 1 ) n + 1 a n {\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}a_{n}} are convergent series. This test is also known as the Leibniz criterion. </p> <h3><a href="/facts/Dirichlet%2527s_test/02ucopM8">Dirichlet's test</a></h3> <p>If { a n } {\displaystyle \{a_{n}\}} is a <a href="/facts/Sequence/gV2S4uqp">sequence</a> of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> and { b n } {\displaystyle \{b_{n}\}} a sequence of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> satisfying </p> <ul><li> a n ≥ a n + 1 {\displaystyle a_{n}\geq a_{n+1}} </li></ul> <ul><li> lim n → ∞ a n = 0 {\displaystyle \lim _{n\rightarrow \infty }a_{n}=0} </li></ul> <ul><li> | ∑ n = 1 N b n | ≤ M {\displaystyle \left|\sum _{n=1}^{N}b_{n}\right|\leq M} for every positive integer <i>N</i></li></ul> <p>where <i>M</i> is some constant, then the series </p> ∑ n = 1 ∞ a n b n {\displaystyle \sum _{n=1}^{\infty }a_{n}b_{n}} <p>converges. </p> <h3><a href="/facts/Cauchy%2527s_convergence_test/FHPAzX0V">Cauchy's convergence test</a></h3> <p>A series ∑ i = 0 ∞ a i {\displaystyle \sum _{i=0}^{\infty }a_{i}} is convergent if and only if for every ε > 0 {\displaystyle \varepsilon >0} there is a natural number <i>N</i> such that </p> | a n + 1 + a n + 2 + ⋯ + a n + p | < ε {\displaystyle |a_{n+1}+a_{n+2}+\cdots +a_{n+p}|<\varepsilon } <p>holds for all <i>n</i> > <i>N</i> and all <i>p</i> ≥ 1. </p> <h3><a href="/facts/Stolz%25E2%2580%2593Ces%25C3%25A0ro_theorem/DNHd7ayz">Stolz–Cesàro theorem</a></h3> <p>Let ( a n ) n ≥ 1 {\displaystyle (a_{n})_{n\geq 1}} and ( b n ) n ≥ 1 {\displaystyle (b_{n})_{n\geq 1}} be two sequences of real numbers. Assume that ( b n ) n ≥ 1 {\displaystyle (b_{n})_{n\geq 1}} is a <a href="/facts/Monotonic_function/MjQK0YL2">strictly monotone</a> and divergent sequence and the following limit exists: </p> lim n → ∞ a n + 1 − a n b n + 1 − b n = l . {\displaystyle \lim _{n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l.\ } <p>Then, the limit </p> lim n → ∞ a n b n = l . {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=l.\ } <h3><a href="/facts/Weierstrass_M-test/MzGjMAFR">Weierstrass M-test</a></h3> <p>Suppose that (<i>f</i><i>n</i>) is a sequence of real- or complex-valued functions defined on a set <i>A</i>, and that there is a sequence of non-negative numbers (<i>M</i><i>n</i>) satisfying the conditions </p> <ul><li> | f n ( x ) | ≤ M n {\displaystyle |f_{n}(x)|\leq M_{n}} for all n ≥ 1 {\displaystyle n\geq 1} and all x ∈ A {\displaystyle x\in A} , and</li> <li> ∑ n = 1 ∞ M n {\displaystyle \sum _{n=1}^{\infty }M_{n}} converges.</li></ul> <p>Then the series </p> ∑ n = 1 ∞ f n ( x ) {\displaystyle \sum _{n=1}^{\infty }f_{n}(x)} <p>converges absolutely and <a href="/facts/Uniform_convergence/ecz9eNWn">uniformly</a> on <i>A</i>. </p> <h3>Extensions to the ratio test</h3> <p>The ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allows one to deal with this case. </p> <h4><a href="/facts/Ratio_test/SpvhbMQF">Raabe–Duhamel's test</a></h4> <p>Let { <i>a</i><i>n</i> } be a sequence of positive numbers. </p><p>Define </p> b n = n ( a n a n + 1 − 1 ) . {\displaystyle b_{n}=n\left({\frac {a_{n}}{a_{n+1}}}-1\right).} <p>If </p> L = lim n → ∞ b n {\displaystyle L=\lim _{n\to \infty }b_{n}} <p>exists there are three possibilities: </p> <ul><li>if <i>L</i> > 1 the series converges (this includes the case <i>L</i> = ∞)</li> <li>if <i>L</i> < 1 the series diverges</li> <li>and if <i>L</i> = 1 the test is inconclusive.</li></ul> <p>An alternative formulation of this test is as follows. Let { <i>a</i><i>n</i> } be a series of real numbers. Then if <i>b</i> > 1 and <i>K</i> (a natural number) exist such that </p> | a n + 1 a n | ≤ 1 − b n {\displaystyle \left|{\frac {a_{n+1}}{a_{n}}}\right|\leq 1-{\frac {b}{n}}} <p>for all <i>n</i> > <i>K</i> then the series {<i>a</i><i>n</i>} is convergent. </p> <h4><a href="/facts/Ratio_test/SpvhbMQF">Bertrand's test</a></h4> <p>Let { <i>a</i><i>n</i> } be a sequence of positive numbers. </p><p>Define </p> b n = ln n ( n ( a n a n + 1 − 1 ) − 1 ) . {\displaystyle b_{n}=\ln n\left(n\left({\frac {a_{n}}{a_{n+1}}}-1\right)-1\right).} <p>If </p> L = lim n → ∞ b n {\displaystyle L=\lim _{n\to \infty }b_{n}} <p>exists, there are three possibilities:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <ul><li>if <i>L</i> > 1 the series converges (this includes the case <i>L</i> = ∞)</li> <li>if <i>L</i> < 1 the series diverges</li> <li>and if <i>L</i> = 1 the test is inconclusive.</li></ul> <h4><a href="/facts/Ratio_test/SpvhbMQF">Gauss's test</a></h4> <p>Let { <i>a</i><i>n</i> } be a sequence of positive numbers. If a n a n + 1 = 1 + α n + O ( 1 / n β ) {\displaystyle {\frac {a_{n}}{a_{n+1}}}=1+{\frac {\alpha }{n}}+O(1/n^{\beta })} for some β > 1, then ∑ a n {\displaystyle \sum a_{n}} converges if α > 1 and diverges if α ≤ 1.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h4><a href="/facts/Ratio_test/SpvhbMQF">Kummer's test</a></h4> <p>Let { <i>a</i><i>n</i> } be a sequence of positive numbers. Then:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>(1) ∑ a n {\displaystyle \sum a_{n}} converges if and only if there is a sequence b n {\displaystyle b_{n}} of positive numbers and a real number <i>c</i> > 0 such that b k ( a k / a k + 1 ) − b k + 1 ≥ c {\displaystyle b_{k}(a_{k}/a_{k+1})-b_{k+1}\geq c} . </p><p>(2) ∑ a n {\displaystyle \sum a_{n}} diverges if and only if there is a sequence b n {\displaystyle b_{n}} of positive numbers such that b k ( a k / a k + 1 ) − b k + 1 ≤ 0 {\displaystyle b_{k}(a_{k}/a_{k+1})-b_{k+1}\leq 0} </p><p>and ∑ 1 / b n {\displaystyle \sum 1/b_{n}} diverges. </p> <h3>Abu-Mostafa's test</h3> <p>Let ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} be an infinite series with real terms and let f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } be any real function such that f ( 1 / n ) = a n {\displaystyle f(1/n)=a_{n}} for all positive integers <i>n</i> and the second derivative f ″ {\displaystyle f''} exists at x = 0 {\displaystyle x=0} . Then ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges absolutely if f ( 0 ) = f ′ ( 0 ) = 0 {\displaystyle f(0)=f'(0)=0} and diverges otherwise.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h3>Notes</h3> <ul><li>For some specific types of series there are more specialized convergence tests, for instance for <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> there is the <a href="/facts/Dini_test/lVH5WDQk">Dini test</a>.</li></ul> <h2 id="examples">Examples</h2> <p>Consider the series </p> <table><tbody><tr><td> ∑ n = 1 ∞ 1 n α . {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{\alpha }}}.} </td> <td></td> <td>i</td></tr></tbody></table> <p><a href="/facts/Cauchy_condensation_test/s7CoyYeY">Cauchy condensation test</a> implies that (i) is finitely convergent if </p> <table><tbody><tr><td> ∑ n = 1 ∞ 2 n ( 1 2 n ) α {\displaystyle \sum _{n=1}^{\infty }2^{n}\left({\frac {1}{2^{n}}}\right)^{\alpha }} </td> <td></td> <td>ii</td></tr></tbody></table> <p>is finitely convergent. Since </p> ∑ n = 1 ∞ 2 n ( 1 2 n ) α = ∑ n = 1 ∞ 2 n − n α = ∑ n = 1 ∞ 2 ( 1 − α ) n {\displaystyle \sum _{n=1}^{\infty }2^{n}\left({\frac {1}{2^{n}}}\right)^{\alpha }=\sum _{n=1}^{\infty }2^{n-n\alpha }=\sum _{n=1}^{\infty }2^{(1-\alpha )n}} <p>(ii) is a geometric series with ratio 2 ( 1 − α ) {\displaystyle 2^{(1-\alpha )}} . (ii) is finitely convergent if its ratio is less than one (namely α > 1 {\displaystyle \alpha >1} ). Thus, (i) is finitely convergent <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> α > 1 {\displaystyle \alpha >1} . </p> <h2 id="convergence-of-products">Convergence of products</h2> <p>While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of <a href="/facts/Infinite_product/qaaNh5PM">infinite products</a>. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be a sequence of positive numbers. Then the infinite product ∏ n = 1 ∞ ( 1 + a n ) {\displaystyle \prod _{n=1}^{\infty }(1+a_{n})} converges <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges. Also similarly, if 0 ≤ a n < 1 {\displaystyle 0\leq a_{n}<1} holds, then ∏ n = 1 ∞ ( 1 − a n ) {\displaystyle \prod _{n=1}^{\infty }(1-a_{n})} approaches a non-zero limit if and only if the series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges . </p><p>This can be proved by taking the logarithm of the product and using limit comparison test.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/L%2527H%25C3%25B4pital%2527s_rule/pCXK75Io">L'Hôpital's rule</a></li> <li><a href="/facts/Shift_rule/dmJDTRAk">Shift rule</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Louis_Leithold/NGV4BHbb">Leithold, Louis</a> (1972). <i>The Calculus, with Analytic Geometry</i> (2nd ed.). New York: Harper & Row. pp. 655–737. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-06-043959-9.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Wachsmuth, Bert G. "MathCS.org - Real Analysis: Ratio Test". www.mathcs.org. <a href="http://www.mathcs.org/analysis/reals/numser/t_ratio.html" target="_blank">http://www.mathcs.org/analysis/reals/numser/t_ratio.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>František Ďuriš, Infinite series: Convergence tests, pp. 24–9. Bachelor's thesis. <a href="http://oldwww.dcs.fmph.uniba.sk/bakalarky/obhajene/getfile.php/new.pdf?id=90&fid=228&type=application%2Fpdf" target="_blank">http://oldwww.dcs.fmph.uniba.sk/bakalarky/obhajene/getfile.php/new.pdf?id=90&fid=228&type=application%2Fpdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Weisstein, Eric W. "Bertrand's Test". mathworld.wolfram.com. Retrieved 2020-04-16. <a href="https://mathworld.wolfram.com/BertrandsTest.html" target="_blank">https://mathworld.wolfram.com/BertrandsTest.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>* "Gauss criterion", Encyclopedia of Mathematics, EMS Press, 2001 [1994] <a href="https://www.encyclopediaofmath.org/index.php?title=Gauss_criterion" target="_blank">https://www.encyclopediaofmath.org/index.php?title=Gauss_criterion</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"Über die Convergenz und Divergenz der unendlichen Reihen". Journal für die reine und angewandte Mathematik. 1835 (13): 171–184. 1835-01-01. doi:10.1515/crll.1835.13.171. ISSN 0075-4102. S2CID 121050774. <a href="https://www.degruyter.com/document/doi/10.1515/crll.1835.13.171/html" target="_blank">https://www.degruyter.com/document/doi/10.1515/crll.1835.13.171/html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Tong, Jingcheng (1994). "Kummer's Test Gives Characterizations for Convergence or Divergence of all Positive Series". The American Mathematical Monthly. 101 (5): 450–452. doi:10.2307/2974907. JSTOR 2974907. <a href="https://www.jstor.org/stable/2974907" target="_blank">https://www.jstor.org/stable/2974907</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Samelson, Hans (1995). "More on Kummer's Test". The American Mathematical Monthly. 102 (9): 817–818. doi:10.1080/00029890.1995.12004667. ISSN 0002-9890. <a href="https://www.tandfonline.com/doi/full/10.1080/00029890.1995.12004667" target="_blank">https://www.tandfonline.com/doi/full/10.1080/00029890.1995.12004667</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Abu-Mostafa, Yaser (1984). "A Differentiation Test for Absolute Convergence" (PDF). Mathematics Magazine. 57 (4): 228–231. doi:10.1080/0025570X.1984.11977116. <a href="https://work.caltech.edu/paper/Abu-Mostafa17280.pdf" target="_blank">https://work.caltech.edu/paper/Abu-Mostafa17280.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Belk, Jim (26 January 2008). "Convergence of Infinite Products". <a href="https://cornellmath.wordpress.com/2008/01/26/convergence-of-infinite-products/" target="_blank">https://cornellmath.wordpress.com/2008/01/26/convergence-of-infinite-products/</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>