3 Sequences and Series

3.5 Series

After having studied the limits of number sequences, we can apply our newly acquired knowledge to topics which occur more often in physics, for instance infinite sums , called series:

These are often encounterered sometimes in more interesting physical questions: For instance if we want to sum up the electrostatic energy of infinitely many equidistant alternating positive and negative point charges for one chain link (which gives a simple but surprisingly good one-dimensional model of a ion crystal) we come across the infinite sum over the members of the alternating harmonic sequence  (F7): the series. How do we calculate this?

Series are sequences whose members are finite sums of real numbers: The definition of a

reduces the series to sequences which we have been dealing with just above.

Especially, a series is exactly then convergent and has the value, if the sequence of its partial sums (not that of its  summands!!) converges: :

Also the multiple of a convergent series and the sum and difference of two convergent series are again convergent.

The few sample series that we need, to see the most important concepts, we derive simply through piecewise summing up our sample sequences:

The series (R1) of the partial sums of the sequence (F1) of the natural numbers: is clearly divergent.

The series (R2) made out of the members of the alternating sequence (F2) always jumps between and , and has therefore two cluster points and consequently no limit.

Also the "harmonic series" (R3), summed up out of the members of the harmonic sequence (F3), i.e. the sequence is divergent. Because the (also necessary) Cauchy Criterion is not fulfilled:
If we for instance choose and consider a piece of the sequence for consisting of terms:
while for convergence would have been necessary.

Their alternating variant (R7) however, created out of the sequence (F7), our physical example from above, converges (, as we will show later). Because of this difference between series with purely positive summands and alternating ones, it is appropriate to introduce a new term: A series is said to be  absolutely convergent, if already the series of the absolute values converges.

We can easily understand that within an absolutely convergent series the summands can be rearranged without any effect on the limiting value. Two absolutely convergent series can be multiplied termwise to create a new absolutely convergent series.

For absolute convergence the mathematicians have developed various sufficient criteria, the so-called majorant criteria, which you will deal with more closely in the lecture about analysis.:

Very often the "geometric series" (R6): , which follow from the geometric sequences (F6) , serve as majorants . To calculate them we benefit from the earlier forderived geometric sum :
meaning convergent for and divergent for .
The series of the inverse natural factorials (R4) deserves to be examined in more detail:

First we realize that the sequence of the partial sums increases monotonically: .

To get an upper bound we estimate through the majorant geometric sum with :

Since the monotonically increasing sequence of the partial sums is bounded from above by, the Theorem of Bolzano and Weierstrass guarantees us convergence. We just do not know the limiting value yet. This limit is indeed something fully new - namely an irrational number. We call it , so that the number after the supplementary convention  is defined by the following series starting with :

 

To get the numerical value of we first calculate the members of the zero sequence (F4) :

then we sum up the partial sums:
If we look at the rapid convergence, we can easily imagine that after a short calculation we receive the following result for the limiting value:

.
Through these considerations we now have got a first overview over the limiting procedures and some of the sequences and series important for natural sciences with their limits, which will be of great use for us in the future.