Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES AND LIMITS

For a series with an infinite number of terms and|r|<1, we have limN→∞rN=0,

and the sum tends to the limit

S=

a

1 −r

. (4.4)

In (4.1),r=^12 ,a=^12 ,andsoS=1.For|r|≥1, however, the series either diverges

or oscillates.

Consider a ball that drops from a height of27 mand on each bounce retains only a third

of its kinetic energy; thus after one bounce it will return to a height of9m,aftertwo

bounces to3m, and so on. Find the total distance travelled between the first bounce and

theMth bounce.

The total distance travelled between the first bounce and theMth bounce is given by the
sum ofM−1 terms:

SM− 1 =2(9+3+1+···)=2

M∑− 2

m=0

9

3 m

forM>1, where the factor 2 is included to allow for both the upward and the downward
journey. Inside the parentheses we clearly have a geometric series with first term 9 and
common ratio 1/3 and hence the distance is given by (4.3), i.e.

SM− 1 =2×

9

[

1 −

( 1

3

)M− 1 ]

1 −^13

=27

[

1 −

( 1

3

)M− 1 ]

,

where the number of termsNin (4.3) has been replaced byM−1.

4.2.3 Arithmetico-geometric series

An arithmetico-geometric series, as its name suggests, is a combined arithmetic

and geometric series. It has the general form

SN=a+(a+d)r+(a+2d)r^2 +···+[a+(N−1)d]rN−^1 =

N∑− 1

n=0

(a+nd)rn,

and can be summed, in a similar way to a pure geometric series, by multiplying

byrand subtracting the result from the original series to obtain

(1−r)SN=a+rd+r^2 d+···+rN−^1 d−[a+(N−1)d]rN.

Using the expression for the sum of a geometric series (4.3) and rearranging, we

find

SN=

a−[a+(N−1)d]rN

1 −r

+

rd(1−rN−^1 )

(1−r)^2

.

For an infinite series with|r|<1, limN→∞rN= 0 as in the previous subsection,

and the sum tends to the limit

S=

a

1 −r

+

rd

(1−r)^2

. (4.5)

As for a geometric series, if|r|≥1 then the series either diverges or oscillates.