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The divergence tests

Let

(1)

be an infinite series .

If the series converges then according to the definition the sequence

(2)

of its partial sums is convergent, say . Then and . This gives the following result (for another proof visit ):

Theorem: If  (1)  is a convergent infinite series, then  .

Or equivalently, we get

The divergence test: If  (1)  is an infinite series such that then the series diverges.

The following extension of the divergence test goes back to L.Olivier [1]:

Theorem: If the series (1) is convergent and its terms form a decreasing sequence of positive numbers, then even .

The proof follows from Cauchy-Bolzano convergence criterion : Given an , there exists such that  

(3)

for every and every . Take and , where denotes the greatest integer function. Then and . Since is decreasing, the left hand side of (3) can be estimated from below by , which implies that , as desired.

Simple example show that the assumption that is decreasing cannot be omitted ( [2]  , Example 1):  Define

Then ,  but ,  for if is a square.

The most natural application of this result is to show that the harmonic series is divergent . However more can be derived using Olivier’s test: To formulate this we need the notion of the asymptotic density.  If is a subset of the set of positive integers, then the asymptotic density of is defined by

if the limit on the right-hand side exists. We also have ( [3]  , Theorem 11.1) :  If exists then

The next result shows the relation between convergence of subseries of the harmonic series and the asymptotic density (for more details consult  [4] and  [5] ):

If is convergent then .

Note that the above theorem stems from [1] , p.34, where Olivier writes:

Donc si l’on trouve, que dans une série infinie, le produit du terms, ou du des groupes de termes qui conservent le même signe, par est zéro, pour , on peut regarder cette seule circonstance comme une marque, que la série est convergente, si le produit n’est pas nul pour .

This is an English translation according to  [6]  :

Therefore if one finds an infinite series, the product of the th term, or of the th group of terms which keep the same sign, by , is zero, for , one can regard precisely this situation as an indicator that the series is convergent, and conversely, the series is not convergent if the product is nonzero for .

That this result in this formulation is not correct was immediately pointed out by N.Abel [7] .

References

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