Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

SERIES AND LIMITS


Sum the series
∑N

n=1

(n+1)(n+3).

Thenth term in this series is


un=(n+1)(n+3)=n^2 +4n+3,

and therefore we can write


∑N

n=1

(n+1)(n+3)=

∑N


n=1

(n^2 +4n+3)

=


∑N


n=1

n^2 +4

∑N


n=1

n+

∑N


n=1

3


=^16 N(N+ 1)(2N+1)+4×^12 N(N+1)+3N


=^16 N(2N^2 +15N+ 31).


4.2.6 Transformation of series

A complicated series may sometimes be summed by transforming it into a


familiar series for which we already know the sum, perhaps a geometric series


or the Maclaurin expansion of a simple function (see subsection 4.6.3). Various


techniques are useful, and deciding which one to use in any given case is a matter


of experience. We now discuss a few of the more common methods.


The differentiation or integration of a series is often useful in transforming an

apparently intractable series into a more familiar one. If we wish to differentiate


or integrate a series that already depends on some variable then we may do so


in a straightforward manner.


Sum the series

S(x)=

x^4
3(0!)

+


x^5
4(1!)

+


x^6
5(2!)

+···.


Dividing both sides byxwe obtain


S(x)
x

=


x^3
3(0!)

+


x^4
4(1!)

+


x^5
5(2!)

+···,


which is easily differentiated to give


d
dx

[


S(x)
x

]


=


x^2
0!

+


x^3
1!

+


x^4
2!

+


x^5
3!

+···.


Recalling the Maclaurin expansion of expxgiven in subsection 4.6.3, we recognise that
the RHS is equal tox^2 expx. Having done so, we can now integrate both sides to obtain


S(x)/x=


x^2 expxdx.
Free download pdf