Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

SERIES AND LIMITS


which is merely the series obtained by settingx= 1 in the Maclaurin expansion of expx
(see subsection 4.6.3), i.e.


exp(1) =e=1+

1


1!


+


1


2!


+


1


3!


+···.


Clearly this second series is convergent, since it consists of only positive terms and has a
finite sum. Thus, since each termunin the series (4.7) is less than the corresponding term
1 /n! in (4.8), we conclude from the comparison test that (4.7) is also convergent.


D’Alembert’s ratio test

The ratio test determines whether a series converges by comparing the relative


magnitude of successive terms. If we consider a series



unand set

ρ= lim
n→∞

(
un+1
un

)
, (4.9)

then ifρ<1 the series is convergent; ifρ>1 the series is divergent; ifρ=1


then the behaviour of the series is undetermined by this test.


To prove this we observe that if the limit (4.9) is less than unity, i.e.ρ<1then

we can find a valuerin the rangeρ<r<1 and a valueNsuch that


un+1
un

<r,

for alln>N. Now the termsunof the series that followuNare


uN+1,uN+2,uN+3, ...,

and each of these is less than the corresponding term of


ruN,r^2 uN,r^3 uN, .... (4.10)

However, the terms of (4.10) are those of a geometric series with a common


ratiorthat is less than unity. This geometric series consequently converges and


therefore, by the comparison test discussed above, so must the original series

un. An analogous argument may be used to prove the divergent case when


ρ>1.


Determine whether the following series converges:
∑∞

n=0

1


n!

=


1


0!


+


1


1!


+


1


2!


+


1


3!


+···=2+


1


2!


+


1


3!


+···.


As mentioned in the previous example, this series may be obtained by settingx=1inthe
Maclaurin expansion of expx, and hence we know already that it converges and has the
sum exp(1) =e. Nevertheless, we may use the ratio test to confirm that it converges.
Using (4.9), we have


ρ= lim
n→∞

[


n!
(n+1)!

]


= lim
n→∞

(


1


n+1

)


= 0 (4.11)


and sinceρ<1, the series converges, as expected.

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