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[[File:Integral Test.svg|thumb|right|300px|The integral test applied to the [[harmonic series (mathematics)|harmonic series]].  Since the area under the curve {{math|''y'' {{=}} 1/''x''}} for {{math|''x'' ∈ {{closed-open|1, ∞}}}} is infinite, the total area of the rectangles must be infinite as well.]]
{{Calculus |Series}}


In [[mathematics]], the '''integral test for convergence''' is a [[convergence tests|method used to test]] infinite [[series (mathematics)|series]] of [[non-negative]] terms for [[convergent series|convergence]]. It was developed by [[Colin Maclaurin]] and [[Augustin-Louis Cauchy]] and is sometimes known as the Maclaurin–Cauchy test.


==Statement of the test==
[http://en.wikipedia.org/wiki/Dog_tag_(identifier) wikipedia.org]Barse Jewelry is really an set up rings corporation operating out of Texas, developed and Dallas within their factory in Thailand even though also you can found their products in retail store options from main shops to retail storefronts in Fort and Southlake Truly worth, TX. The innovator and founder themselves, Melanie Barse, layouts every bit of these lovely jewelry pieces. She actually is the innovative mind associated with a crew of qualified fashion designers that provide and generate refreshing selections alive.<br><br>Together with her intrepid eyes and style for necklaces and also her razor-sharp business intuition, this company rapidly became to become prosperous global import business that helps to keep making modern and stylish components. Every single layout features its own one of a kind and distinctive impression of Barse Jewelry having verified Melanie’s appreciate and expertise of facts. She themselves arrived downward with guidelines on the miners and carverscarvers and artisans of methods to design a natural stone in a distinguished way in which will make her designs totally different from designers who use pre-slice rocks.<br><br>The jewelry pieces use sterlingbronze and gold, copper or bronze with gold overlay. They all are designed using the same artisan procedure for hands wax tart throwing and nickel free of charge. The stones applied are mainly honest see through, opaque and rocks gemstones that indicate present coloration developments, and as well glass or crystal to increase added shimmer.<br><br>Barse Jewelry - The Rewards<br><br>Almost all of the metals found in Barse Jewelry is silver so that it is a kind of good expenditure themselves as the value of silver has rocketed up within the past a long time.<br><br>It is just a trusted and trusted web page that provides its shoppers and consumers with a large variety of actually wonderful expensive jewelry you in no way see in other places.<br><br>It truly is your go-to-retail outlet which you could retail outlet for your very own selections or products.<br><br>The quality is without a doubt of your premier along with the best price in town.<br><br>Secure and safe shopping knowledge. You do not have to worry about your information staying exposed to the third bash.<br><br>There are various selections in delivery your sales with wide-ranging costs.<br><br>There is also promise within one month of invoice if any damages or discrepancies, you could go back your package returning to Barse.<br><br>Support services crew job very well with  [http://barsejewelryreview.com/ barse jewelry outlet] good services and there is phone number on the market if you ever require further services or info.<br><br>Barse Jewelry - Net Profit<br><br>Uncover all of the favored jewelry pieces using your most preferred patterns and forms within one put. You may go shopping by category or search for buyers’ preferred. Regardless of whether you wish to search for stunning part in silver as well as their surprising bronze selection and honest rocks, such as opal, amethyst, onyx, pearl and the very simple yet stylish turquoise. All immaculately linked to the selection of pendants, ear-rings, jewelry, a lot of, pendants and charms even more. Be feel comfortable knowing that you can enjoy a satisfying searching retailer and expertise on the internet with obtain just with Barse Jewelry.
Consider an [[integer]] {{math|''N''}} and a non-negative function {{math|''f''}} defined on the unbounded [[interval (mathematics)|interval]] {{closed-open|''N'', ∞}}, on which it is [[monotone decreasing]]. Then the infinite series
 
:<math>\sum_{n=N}^\infty f(n)</math>
 
converges to a [[real number]] if and only if the [[improper integral]]
 
:<math>\int_N^\infty f(x)\,dx</math>
 
is finite. In other words, if the integral diverges, then the [[divergent series|series diverges]] as well.
 
===Remark===
The proof also gives the [[upper and lower bounds|lower and upper bounds]]
 
:<math>\int_N^\infty f(x)\,dx\le\sum_{n=N}^\infty f(n)\le f(N)+\int_N^\infty f(x)\,dx</math> 
 
for the infinite series.
 
==Proof==
The proof basically uses the [[comparison test]], comparing the term {{math|''f''(''n'')}} with the integral of {{math|''f''}} over the intervals
{{closed-open|''n'' − 1, ''n''}} and {{closed-open|''n'', ''n'' + 1}}, respectively.
 
Since {{math|''f''}} is a monotone decreasing function, we know that
 
:<math>
f(x)\le f(n)\quad\text{for all }x\in[n,\infty)
</math>
 
and
 
:<math>
f(n)\le f(x)\quad\text{for all }x\in[N,n].
</math>
 
Hence, for every integer {{math|''n'' ≥ ''N''}},
 
{{NumBlk|:|<math>
\int_n^{n+1} f(x)\,dx
\le\int_{n}^{n+1} f(n)\,dx
=f(n)</math>|{{EquationRef|1}}}}
 
and, for every integer {{math|''n'' ≥ ''N'' + 1}},
 
{{NumBlk|:|<math>
f(n)=\int_{n-1}^{n} f(n)\,dx
\le\int_{n-1}^n f(x)\,dx.
</math>|{{EquationRef|2}}}}
 
By summation over all {{math|''n''}} from {{math|''N''}} to some larger integer {{math|''M''}}, we get from ({{EquationNote|1}})
 
:<math>
\int_N^{M+1}f(x)\,dx=\sum_{n=N}^M\underbrace{\int_n^{n+1}f(x)\,dx}_{\le\,f(n)}\le\sum_{n=N}^Mf(n)
</math>
 
and from ({{EquationNote|2}})
 
:<math>
\sum_{n=N}^Mf(n)\le f(N)+\sum_{n=N+1}^M\underbrace{\int_{n-1}^n f(x)\,dx}_{\ge\,f(n)}=f(N)+\int_N^M f(x)\,dx.
</math>
 
Combining these two estimates yields
 
:<math>\int_N^{M+1}f(x)\,dx\le\sum_{n=N}^Mf(n)\le f(N)+\int_N^M f(x)\,dx.</math> 
 
Letting {{math|''M''}} tend to infinity, the result follows.
 
==Applications==
The [[harmonic series (mathematics)|harmonic series]]
:<math>
\sum_{n=1}^\infty \frac1n
</math>
diverges because, using the [[natural logarithm]], its [[derivative]], and the [[fundamental theorem of calculus]], we get
:<math>
\int_1^M\frac1x\,dx=\ln x\Bigr|_1^M=\ln M\to\infty
\quad\text{for }M\to\infty.
</math>
Contrary, the series
:<math>
\sum_{n=1}^\infty \frac1{n^{1+\varepsilon}}
</math>
(cf. [[Riemann zeta function]])
converges for every {{math|''ε'' > 0}}, because by the [[power rule]]
:<math>
\int_1^M\frac1{x^{1+\varepsilon}}\,dx
=-\frac1{\varepsilon x^\varepsilon}\biggr|_1^M=
\frac1\varepsilon\Bigl(1-\frac1{M^\varepsilon}\Bigr)
\le\frac1\varepsilon<\infty
\quad\text{for all }M\ge1.
</math>
 
==Borderline between divergence and convergence==
The above examples involving the harmonic series raise the question, whether there are monotone sequences such that {{math|''f''(''n'')}} decreases to 0 faster than {{math|1/''n''}} but slower than {{math|1/''n''<sup>1+''ε''</sup>}} in the sense that
:<math>
\lim_{n\to\infty}\frac{f(n)}{1/n}=0
\quad\text{and}\quad
\lim_{n\to\infty}\frac{f(n)}{1/n^{1+\varepsilon}}=\infty
</math>
for every {{math|''ε'' > 0}}, and whether the corresponding series of the {{math|''f''(''n'')}} still diverges. Once such a sequence is found, a similar question can be asked with {{math|''f''(''n'')}} taking the role of {{math|1/''n''}}, and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series.
 
Using the integral test for convergence, one can show (see below) that, for every [[natural number]] {{math|''k''}}, the series
{{NumBlk|:|<math>
\sum_{n=N_k}^\infty\frac1{n\ln(n)\ln_2(n)\cdots \ln_{k-1}(n)\ln_k(n)}
</math>|{{EquationRef|3}}}}
still diverges (cf. [[proof that the sum of the reciprocals of the primes diverges]] for {{math|''k'' {{=}} 1}}) but
{{NumBlk|:|<math>
\sum_{n=N_k}^\infty\frac1{n\ln(n)\ln_2(n)\cdots\ln_{k-1}(n)(\ln_k(n))^{1+\varepsilon}}
</math>|{{EquationRef|4}}}}
converges for every {{math|''ε'' > 0}}. Here {{math|ln<sub>''k''</sub>}} denotes the {{math|''k''}}-fold [[function composition|composition]] of the natural logarithm defined [[recursion|recursively]] by
:<math>
\ln_k(x)=
\begin{cases}
\ln(x)&\text{for }k=1,\\
\ln(\ln_{k-1}(x))&\text{for }k\ge2.
\end{cases}
</math>
Furthermore, {{math|''N''<sub>''k''</sub>}} denotes the smallest natural number such that the {{math|''k''}}-fold composition is well-defined and {{math|ln<sub>''k''</sub>(''N''<sub>''k''</sub>) ≥ 1}}, i.e.
:<math>
N_k\ge \underbrace{e^{e^{\cdot^{\cdot^{e}}}}}_{k\ e'\text{s}}=e \uparrow\uparrow k
</math>
using [[tetration]] or [[Knuth's up-arrow notation]].
 
To see the divergence of the series ({{EquationNote|3}}) using the integral test, note that by repeated application of the [[chain rule]]
:<math>
\frac{d}{dx}\ln_{k+1}(x)
=\frac{d}{dx}\ln(\ln_k(x))
=\frac1{\ln_k(x)}\frac{d}{dx}\ln_k(x)
=\cdots
=\frac1{x\ln(x)\cdots\ln_k(x)},
</math>
hence
:<math>
\int_{N_k}^\infty\frac{dx}{x\ln(x)\cdots\ln_k(x)}
=\ln_{k+1}(x)\bigr|_{N_k}^\infty=\infty.
</math>
To see the convergence of the series ({{EquationNote|4}}), note that by the [[power rule]], the chain rule and the above result
:<math>
-\frac{d}{dx}\frac1{\varepsilon(\ln_k(x))^\varepsilon}
=\frac1{(\ln_k(x))^{1+\varepsilon}}\frac{d}{dx}\ln_k(x)
=\cdots
=\frac{1}{x\ln(x)\cdots\ln_{k-1}(x)(\ln_k(x))^{1+\varepsilon}},
</math>
hence
:<math>
\int_{N_k}^\infty\frac{dx}{x\ln(x)\cdots\ln_{k-1}(x)(\ln_k(x))^{1+\varepsilon}}
=-\frac1{\varepsilon(\ln_k(x))^\varepsilon}\biggr|_{N_k}^\infty<\infty.
</math>
 
==References==
* Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.3) ISBN 0-486-60153-6
* [[Whittaker and Watson|Whittaker, E. T., and Watson, G. N., ''A Course in Modern Analysis'']], fourth edition, Cambridge University Press, 1963. (§ 4.43) ISBN 0-521-58807-3
 
<references/>
 
{{DEFAULTSORT:Integral Test For Convergence}}
[[Category:Integral calculus]]
[[Category:Mathematical series]]
[[Category:Convergence tests]]
[[Category:Articles containing proofs]]

Latest revision as of 08:21, 12 October 2014


wikipedia.orgBarse Jewelry is really an set up rings corporation operating out of Texas, developed and Dallas within their factory in Thailand even though also you can found their products in retail store options from main shops to retail storefronts in Fort and Southlake Truly worth, TX. The innovator and founder themselves, Melanie Barse, layouts every bit of these lovely jewelry pieces. She actually is the innovative mind associated with a crew of qualified fashion designers that provide and generate refreshing selections alive.

Together with her intrepid eyes and style for necklaces and also her razor-sharp business intuition, this company rapidly became to become prosperous global import business that helps to keep making modern and stylish components. Every single layout features its own one of a kind and distinctive impression of Barse Jewelry having verified Melanie’s appreciate and expertise of facts. She themselves arrived downward with guidelines on the miners and carverscarvers and artisans of methods to design a natural stone in a distinguished way in which will make her designs totally different from designers who use pre-slice rocks.

The jewelry pieces use sterlingbronze and gold, copper or bronze with gold overlay. They all are designed using the same artisan procedure for hands wax tart throwing and nickel free of charge. The stones applied are mainly honest see through, opaque and rocks gemstones that indicate present coloration developments, and as well glass or crystal to increase added shimmer.

Barse Jewelry - The Rewards

Almost all of the metals found in Barse Jewelry is silver so that it is a kind of good expenditure themselves as the value of silver has rocketed up within the past a long time.

It is just a trusted and trusted web page that provides its shoppers and consumers with a large variety of actually wonderful expensive jewelry you in no way see in other places.

It truly is your go-to-retail outlet which you could retail outlet for your very own selections or products.

The quality is without a doubt of your premier along with the best price in town.

Secure and safe shopping knowledge. You do not have to worry about your information staying exposed to the third bash.

There are various selections in delivery your sales with wide-ranging costs.

There is also promise within one month of invoice if any damages or discrepancies, you could go back your package returning to Barse.

Support services crew job very well with barse jewelry outlet good services and there is phone number on the market if you ever require further services or info.

Barse Jewelry - Net Profit

Uncover all of the favored jewelry pieces using your most preferred patterns and forms within one put. You may go shopping by category or search for buyers’ preferred. Regardless of whether you wish to search for stunning part in silver as well as their surprising bronze selection and honest rocks, such as opal, amethyst, onyx, pearl and the very simple yet stylish turquoise. All immaculately linked to the selection of pendants, ear-rings, jewelry, a lot of, pendants and charms even more. Be feel comfortable knowing that you can enjoy a satisfying searching retailer and expertise on the internet with obtain just with Barse Jewelry.

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