Comparison series-integral

The series S are a process of summation of discrete sizes, the Intégrale of continuous sizes. The formal analogy between the two fields makes it possible to make pass from the interesting ideas of the one to the other. The explicit comparison of an associated integral and a series makes it possible for example to use one to have approximate values of the other.

Formal comparison

From the numerical series of general term u_n, one manufactures a function continues per pieces F , defined by F (X) =u_n for X in

Then the integral of F on $\backslash \; mathbb\; \{R\}\; ^+$ and the series are of comparable nature (both convergent, or both divergent).

In this direction the theory of the series can be seen like a particular case of the study of the convergence of the integrals in the vicinity of $+\; \backslash \; infty$.

It is necessary to however take guard which the integrals conceal a range of behaviors richer than the series, thus

• it is known that if the series of general term u_n converges, then the continuation of general term u_n tends towards 0
• a contrario, there exist functions F of convergent integral (even absolutely convergent) and such as F does not tend towards 0. It is the case of the Intégrale of Fresnel for example.

Theorem of comparison

One supposes this time that the series to be expressed in a form clarifies u_n= F (N) . Of course if F “changes too much” between two consecutive whole values, there is no reason that there is bond between series and integral.

One will thus add assumptions of behavior on F to obtain positive results of comparison.

For monotonous functions

Basic principle

If F is decreasing and continuous on the interval $\backslash \; infty\; [$, then one can frame : $\backslash \; forall\; T\; \backslash \; in,\; \backslash \; qquad\; F\; (n+1)\; \backslash \; Leq\; F\; (T)\; \backslash \; Leq\; F\; (N)\; \backslash \; qquad\; \backslash \; hbox\; \{then\}\; F\; (n+1)\; \backslash \; Leq\; \backslash \; int\_n^\; \{n+1\}\; F\; (T)\; dt\; \backslash \; Leq\; F\; (N)$ Framing which one can reverse in a framing of u_n

$\backslash \; forall\; N\; >2,\; \backslash \; qquad\; \backslash \; int\_n^\; \{n+1\}\; F\; (T)\; dt\; \backslash \; Leq\; u\_n\; \backslash \; Leq\; \backslash \; int\_\; \{n-1\}\; ^\; \{N\}\; F\; (T)\; dt$

One can summon these framings in order to obtain

• a framing of the continuation of the partial sums (attention with the first term)

$\backslash \; int\_0^\; \{N+1\}\; F\; (T)\; dt\; \backslash \; Leq\; \backslash \; sum\_\; \{n=0\}\; ^N\; u\_n\; \backslash \; Leq\; u\_0+\; \backslash \; int\_\; \{0\}\; ^\; \{NR\}\; F\; (T)\; dt$

This framing can give the Limite or an equivalent for the continuation of the partial sums.

• the theorem of comparison

If $f\; \backslash ,$ is a positive function decreasing continuous on the interval $\backslash \; infty\; [$, then the series $\backslash \; sum\; F\; (N)$ and it [[integral] $\backslash \; int\_\; \{0\}\; ^\; \{\backslash \; infty\}\; F\; (X)\; \backslash ,\; dx$ are of comparable nature, i.e. the series is convergent if and only if the integral is convergent.

• In the event of convergence, a framing of the continuation of the remainders

$\backslash \; int\_\; \{N+1\}\; ^\; \{+\; \backslash \; infty\}\; F\; (T)\; dt\; \backslash \; Leq\; \backslash \; sum\_\; \{n=N+1\}\; ^\; \{+\; \backslash \; infty\}\; u\_n\; \backslash \; Leq\; \backslash \; int\_\; \{NR\}\; ^\; \{+\; \backslash \; infty\}\; F\; (T)\; dt$

Again, that can give an equivalent for the continuation of the remainders.

Asymptotic formulation

The preceding framings make it possible to obtain better than a simple equivalent: an asymptotic relation. One can quote celebrates it formula of Euler (which relates to the harmonic Série) as example

$\backslash \; sum\_\; \{n=1\}\; ^N\; \backslash \; frac1n=\; \backslash \; ln\; n+\; \backslash \; gamma+o\; (1)$

What follows explains how to obtain it, and generalize the study with other series.

One replaces on the assumptions of the theorem of comparison integral series above, but one takes the bull by the horns by studying the difference

$\backslash \; Delta\_n\; =\; u\_n-\; \backslash \; int\_n^\; \{n+1\}\; F\; (T)\; dt$

This one thus checks the framing

$0\; \backslash \; Leq\; \backslash \; Delta\_n\; \backslash \; Leq\; u\_n-u\_\; \{n+1\}$

What shows that the series of general term $\backslash \; Delta\_n$ with positive terms and is raised by a series with telescopic terms, convergent. Thus the series of general term $\backslash \; Delta\_n$ converges. One can thus write

$\backslash \; sum\_\; \{n=0\}\; ^N\; u\_n=\; \backslash \; int\_0^\; \{N+1\}\; F\; (T)\; dt+\; \backslash \; sum\_\; \{n=0\}\; ^N\; \backslash \; Delta\_n=\; \backslash \; int\_0^\; \{N+1\}\; F\; (T)\; dt+\; \backslash \; Delta\; +\; O\; (1)$

Continuation of the asymptotic development

One was satisfied to say that the series of general term Δ_n converged. To go further, and to estimate his speed of convergence, one can apply to this same series the method of comparison integral series: we need initially an equivalent for Δ_n

$\backslash \; Delta\_n\; =\; \backslash \; frac1n-\; \backslash \; ln\; (n+1)\; +\; \backslash \; ln\; N\; =\; \backslash \; frac1n-\; \backslash \; ln\; \backslash \; left\; (1+\; \backslash \; frac1n\; \backslash \; right)\; \backslash \; sim\; \backslash \; frac1\; \{2n^2\}$

One then compares the remainder of the series of general term Δ_n with the integral of the function $t\; \backslash \; mapsto\; \backslash \; frac1\; \{2t^2\}$ which is still continuous positive decreasing

$\backslash \; int\_\; \{NR\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{dt\}\; \{2t^2\}\; \backslash \; Leq\; \backslash \; sum\_\; \{n=N+1\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; Delta\_n\; =\; \backslash \; Delta-\; \backslash \; Delta\; \_N\; \backslash \; Leq\; \backslash \; int\_\; \{N+1\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{dt\}\; \{2t^2\}$

What gives a development of Δ_n that one can defer in the formula of Euler. One can start again the operation carried out then, by again withdrawing the integral with which one has just made the comparison. The method continues until obtaining a development with the desired order. For example

$\backslash \; sum\_\; \{k=1\}\; ^n\; \backslash \; frac1k=\; \backslash \; ln\; (N)\; +\; \backslash \; gamma+\; \backslash \; frac1\; \{2n\}\; -\; \backslash \; frac1\; \{12n^2\}\; +\; \backslash \; frac1\; \{120n^4\}\; -\; \backslash \; frac1\; \{252n^6\}\; +\; \backslash \; frac1\; \{240n^8\}\; -\; \backslash \; frac1\; \{132n^\; \{10\}\}\; +$

O \ left (\ frac1 {n^ {12}} \ right)

Alternatives

The comparison series-integral can give fruit even if all the assumptions of the theorem of comparison above are not joined together. One will form the same series Δ_n and one will have to try to make the study of it.

A possible idea, if F is sufficiently regular, is to write Δ_n in the following form (by Intégration by parts)

$\backslash \; Delta\_n\; =\; \backslash \; int\_n^\; \{n+1\}\; (t-n)\; f\text{'}\; (T)\; dt$

For example if the function f' is integrable, one can obtain a result. Intuitively, success is related to the fact that F varies little on.

However a method often more fertile is to proceed directly on the series of general term u_n by applying a to him Transformation of Abel, who is the discrete analog of integration by parts. We present this analogy in the next paragraph.

One can also often apply powerful the Formule sommatoire of Abel.

Derived, primitive, IP

One can proceed in the way of the analogy series-integral. Without claim to provide a rigorous statement, it can be good to regard the following operations as “analogues in a certain direction”. That can guide in the study of problems of analysis.

Example: the series of general term $\backslash \; frac\; \{\backslash \; sin\; N\}\; \{N\}$

The series is not with positive terms, the traditional criteria hardly help us. But by analogy with the study of the convergence of $\backslash \; int\_0^\; \{+\; \backslash \; infty\}\; \backslash \; frac\; \{\backslash \; sin\; T\}\; \{T\}\; dt$ (which is done by integration by parts), one proceeds to a transformation of Abel:

$\backslash \; sum\_\; \{n=1\}\; ^N\; (\backslash \; sin\; N).\; \backslash \; frac\; \{1\}\; \{N\}\; =\; \backslash \; frac1\; \{N+1\}\; (\backslash \; sum\_\; \{k=1\}\; ^N\; \backslash \; sin\; K)\; +\; \backslash \; sum\_\; \{n=1\}\; ^N\; \backslash \; left\; (\backslash \; sum\_\; \{k=1\}\; ^n\; \backslash \; sin\; K\; \backslash \; right)\; \backslash \; left\; (\backslash \; frac1\; \{N\}\; -\; \backslash \; frac1\; \{n+1\}\; \backslash \; right)$

It will be noted that one finds even the “terms between hook” in integration by parts. It remains to apply a trigonometrical Identité (see more precisely Noyau of Dirichlet) to show than the continuation of general term $\backslash \; sum\_\; \{k=1\}\; ^N\; \backslash \; sin\; k$ is limited. Then the theorems of comparison apply and it is obtained that the series of general term $\backslash \; frac\; \{\backslash \; sin\; N\}\; \{N\}$ converges.

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