4.2 Limits

Limit

What happens if the index of the sequence or series becomes larger and larger, i.e. if it goes to infinity. Are the terms of the sequence getting larger and larger (in this case we call the sequence divergent), or do they approach a limiting values, i.e. the sequence is convergent? Does the value of the series grows to infinity or does it remain bounded, i.e. does it have a limit and is convergent?

The sequence of the natural numbers obviously grows without limit as well as the value of the series; both are divergent:

limnAn = limnn = limnSn = limn m=1nm =

What about the geometric series?

limnAn = limnan = 0for a < 0 = 1fora = 1 = for a > 1 no limitfora -1 limnSn = limn m=0nam = 1 1-afor a < 1 fora 1  no limit for a - 1

For a > 1 the terms of the geometric sequence grow continuously , thus neither the sequence not the resulting series have a finite limit. For a = 1 the terms of the sequence are constant; the partial sums of the series correspond to the sequence of the natural numbers and thus the series is divergent.

For 0 < a < 1 the terms of the sequence are getting smaller and smaller and their limit is zero. The series converges to the limit 1(1 - a), which is larger than 1.

For - 1 < a < 0 the terms of the sequence are getting smaller and smaller while changing sign and the series is convergent with the limit 1(1 + |a|), which is smaller than 1.

For a = -1 the sequence alternates between 1 and - 1 and the partial sums (1 - 1 + 1 - 1) is either 1 or 0 depending on the index. Therefore no limit exists. For a < -1 the terms of the sequence ,as well as the partial sums have alternating signs while growing in absolute value. Their absolute values go to infinity. Therefore the sequence and series itself do not have a limit.

The following simulation shows the behaviour of the geometric sequence and series as function of the parameter a, which can be adjusted with a slider.


PIC


Figure 4.1: The first window shows the terms of the geometric sequence, the second window the partial sums of the geometric series as a function of N, with the red line as limit, provided the limit exists within the shown range of ordinates.


Figure 4.1b not translated yet!!


Figure 4.2: The third window shows the limit of the series as a function of a for |a| < 1. The red point marks the value of a chosen with the slide control.

What are the conditions for a series in order for it to have a limit? Obviously the terms of the associated sequence must converge to 0. That is a necessary, but not yet a sufficient condition. An example to illustrate the difference is the harmonic series:

harmonic seriesA = 1,1 2,1 3,1 4,1 5,1 6,... A1 = 1;An = 1 n;limnAn = limn1 n = 0 Sn = 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + ..

While the terms of the sequence converge to 0, the series of partial sums grows without limit and thus does not have a limiting value.

This becomes evident in the easiest way, if one compares the harmonic series, with a series, that obviously diverges , and whose suitably grouped terms are smaller or equal than those of the harmonic sequence:

Sharmonic = 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + 1 9 + 1 10 + 1 11 + 1 12 + 1 13 + 1 14 + 1 15 + ... Scomparison = 1 + 1 2 + 1 2 + 1 4 + 1 4 + 1 4 + 1 4 + 1 8 + 1 8 + 1 8 + 1 8 + 1 8 + 1 8 + 1 8 + 1 8 + ... S comparison = 1 + 1 + 1 + 1, +... Sharmonic < Scomparison Sharmonic

Thus, the terms of the harmonic sequence do not converge strongly enough to zero to ensure convergence.

A sufficient criterion for convergence is, that the ratio of successive terms of the sequence is smaller than 1 for n ( quotient criterion of d’Alembert). For the two series we have:

harmonic Series An+1 An = n n+1;limn n n+1 = 1 geometric Series An+1 An = an+1 an = a;limxa = a < 1for a < 1

While the consecutive terms of the geometric sequence decay for a < 1 in a constant proportion for the geometric series, the terms of the harmonic sequence keep on decaying, but in the limit of n consecutive terms are becoming “equal”.