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Maclaurin-Cauchy integral test

This is one of the basic tests given in elementary courses on analysis:

Theorem: Let be a non-negative, decreasing function defined on interval . Then   converges if and only if the improper integral    converges.  

An early form of this was discovered in India  by Madhava of Sangamagramma   in the 14th century.  The basic idea of the proof and application of this result appears in Europe for the first time in Maclaurin’s ¹  Treatise of Fluxions [1] . He considered convergence of series whose terms are given by rational function. He writes in this connection [2]  : When the ares   has a limit, we not only conclude from this, that the sum of the progression represented by the ordinates has a limit; but when the former limit is known, we may by it approximate to the value of the latter ... (Here the area   is the area is .)

The test was later rediscovered by A.Cauchy  [3] .   

The proof is simple. Since for and  , we have   . Consequently

The left hand side inequality  shows the convergence of the integral provided the series is convergent, and the right hand side inequality shows the sufficient condition.

The geometric interpretation of the above proof is: Let , ,  ,  and . Then represents between the green and blue graphs, and between the green and red graphs. Both and increase with . Since the sum equals there follows that each sequences has a positive limit less than .

Note that when the monotonic condition is removed from the assumptions, the theorem is no longer true.

Corollary: The difference converges for to a limit between and .

For the proof observe that if and for , then . Consequently, , and the result follows.

A simple consequence of the Corollary is the existence of the limit

known as the Euler-Mascheroni constant , and its very rough estimate . More generally, if with then there exists the limit   

and the limit lies between and . If for instance,   then

where denotes the Riemann zeta function.

De la Vallée-Poussin [4] gives as an exercise (Dover Publications, New York 1946, p.414) the following result:

If , , , and if is positive monotone decreasing, then converges with , and   diverges with .

Another form of  De la Vallée-Poussin result is: If   is a divergent series of positive terms and is positive monotone decreasing with then converges if is convergent; and that diverges if is divergent, where .

The analytical essence of the integral test is as follows: Let a function be such that . Then the mean value theorem shows that for some . When we “replace” be we get . Consequently, under suitable analytical restrictions   and have the same convergence behavior.

Bromwich  [5]  proved the following generalization:

Theorem: Suppose that with

(α) tends steadily to zero,

(β)   tends steadily to infinity,

(γ) tends steadily to zero.

If moreover

(δ) the integral is convergent ,then

(I) the series and the integral   converge, diverge, or oscillate together.

(II) the series and the integral   converge, diverge, or oscillate together.

(III) the series and the integral   converge, diverge, or oscillate together.

Hardy  [6]   simplified and extended Bromwich’ result proving:

Theorem: Let

(i) possesses a continuous derivative on ,

(ii) ,

(iii) the integral   is convergent.

Then tends to a finite limit as .

R.W.Brink  [7]  further extended Hardy result in several ways. One of his extensions says:

Theorem: Let be an integrable function of real variable such that

(1) ,

(2)   for   , the series being a convergent series.

A necessary and sufficient condition for the convergence of the series is the convergence of the integral .

Du Bois-Reymond  [8]  calls  tests that use the ratio as  tests of the second type to distinguish them from the tests using the general term itself, which he calls tests of the first kind.

A connecting idea between both types of tests is based on the following reasoning:  Given a sequence , let . Then . Suppose that is a positive continuous function such that .  Suppose that   is a solution of the difference equation satisfying .  By the mean value theorem for some . Under reasonable conditions on one can expect that differs from by a negligible error for large values of so that we can write . Applying the original Maclaurin-Cauchy test we are to the integral

R.W.Brink  [9] , [7] proved several integral tests of the second type which embody the integral test. One of them is:

Theorem: Let    be a series of positive terms. Let    be a function such that

(i) ,

(ii) ,

(iii) exists and is continuos, is convergent.

Then the convergence of the integral   is necessary and sufficient for the convergence of the series .

For Brink’s result follows form the following Rajagopal’s one  [10]

Theorem:  Let be a series of positive terms. If

(i) is a strictly increasing sequence tending to infinity;

(ii) ;

(iii) has a continuous derivative and is convergent;

(iv)     (C):      is convergent,  or    (D):       is divergent;

(v)     (C):    ,   or    (D):    ;

then is  (C)  convergent,  or  (D)  divergent.

There is no end with possible generalizations of the original Maclaurin-Cauchy test. One possible way to go is the following  [11] : If is Riemann integrable on a finite interval then the Riemann sums

converges to for .  

Maclaurin-Cauchy test can also be extended to double series:

Theorem: If the function is a positive and steadily decreases to zero as and increase to infinity (that is, if and ), then the double series converges or diverges with the double interval .

Notes

References

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