Understanding Engineering Mathematics

(やまだぃちぅ) #1

D’Alembert’s ratio test


Lest, as an engineer, you doubt the relevance of all this pure mathematics to your studies,
a few words about Jean Le Rond D’Alembert (1717 – 1783). Thought to be the illegitimate
son of a Parisian noble who paid for his education, he was found as a baby on the steps of
the church of St Jean Le Rond. He is most famous for his work in the field of dynamics
(d’Alembert’s Principle) where he made great contributions to applying Newton’s Laws
to the motion of complicated systems of planets. In this he helped lay the foundations
for the approximation methods now used routinely in the analysis of complex mechanical
systems occurring in all aspects of engineering. His convergence test described here is just
one small example of the sorts of mathematical tool one needs to apply and justify such
approximation methods.
This useful test, thed’Alembert ratio test, deals with the positive values of the ratio
of successive terms.
For the series


S=u 1 +u 2 +···+un+···

let


lim
n→∞





un+ 1
un




∣=l

Then the series isdivergent ifl> 1
convergent ifl< 1
and ifl=1 the test fails and we have to find some other test, for example, the compar-
ison test.


We can see roughly why the ratio test works. We note that limn→∞





un+ 1
un




∣gives the

behaviour of the ‘last terms of the series’ – after a large number of terms the series will
have terms∣ un,un+ 1 ...which we can compare with a geometric series with common ratio



un+ 1
un




∣. If the common ratio is less than 1 the series will converge, otherwise it will diverge

(unless possibly it is equal to 1, in which case further investigation is needed).


Problem 14.14
Examine the convergence of the following series by the ratio test:

(i) 1Y

1
2!

Y

1
3!

Y

1
4!

Y··· (ii) 1Y

2
1

Y

22
2!

Y

23
3!

Y

24
4!

Y

25
5!

Y···

(iii) 1Y

1
22

Y

1
32

Y

1
42

Y

1
52

Y···

(i) We have
un=

1
n!

,un+ 1 =

1
(n+ 1 )!

So
lim
n→∞





un+ 1
un




∣=nlim→∞





1
(n+ 1 )!

n!




∣=^0 <^1
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