arithmetic operations such as addition and multiplication. Fortunately, we can express
cosx(xin radians!) in terms of just such operations by aninfinite power series(107
➤
):
cosx= 1 −x^2
2!+x^4
4!−x^6
6!+···If this is new to you, don’t worry where it comes from just now. Because the series is
infinite – the terms go on forever – we can’t even total all of the terms and so can never
write down the exact value of, for example, cos 0.1:
cos 0. 1 = 1 −( 0. 1 )^2
2!+( 0. 1 )^4
4!−( 0. 1 )^6
6!+···However, by taking a sufficient number of terms we can obtain the value of cos 0.1 with
as much accuracy as we desire. The terms in the series only involve multiplication and
division, which a computer can do easily. So such series are just the thing for evaluating
complicated functions. They are also useful in mathematical methods too – for example
if x is small enough then we might be able to neglect terms such asx^4 ,x^6 ,...and take
theapproximation:
cosx 1 −x^2
2The advantage here is that the right-hand side is much easier to work with than the
left-hand side.
We will look at how we find such series later, in Section 14.11. Here we will focus on
issues arising from the fact that we have aninfiniteseries. Because it is infinite, we can
never see directly what is going to happen when we ‘gather up’ all the terms. For the cosx
example above, the terms
( 0. 1 )r
r!got smaller and smaller, so maybe we can feel confidentthat we are all right to neglect the remaining infinity of terms after we cut the series off.
Well, look at another series,
S= 1 +^12 +^13 +^14 +···The terms get smaller and smaller – maybe we can get a good approximation by ‘trun-
cating’ the series? In fact, we can’t. This can be seen by a very pretty argument. BecauseS
is the sum of only positive terms, we can obviously get a quantitylessthanSby replacing
some of its terms by smaller values. Suppose we did this as follows
S> 1 +^12 +( 1
4 +1
4)
+( 1
8 +1
8 +1
8 +1
8)
+···You can see the pattern:
= 1 +^12 +^12 +^12 +···But this clearly adds up to an infinite total and yet we know by construction that this is
less thanS–soSmust also be infinite, showing that the series diverges.
This is typical of the sort of ‘strange’ results and pretty arguments one meets in the
theory of series. From a practical point of view it means we have to be very careful when
dealing with infinite series.