Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES AND LIMITS

Again using the Maclaurin expansion of expxgiven in subsection 4.6.3, we notice that

S(θ) = Re [exp(expiθ)] = Re [exp(cosθ+isinθ)]

=Re{[exp(cosθ)][exp(isinθ)]}= [exp(cosθ)]Re [exp(isinθ)]

=[exp(cosθ)][cos(sinθ)].

4.3 Convergence of infinite series

Although the sums of some commonly occurring infinite series may be found,

the sum of a general infinite series is usually difficult to calculate. Nevertheless,

it is often useful to know whether the partial sum of such a series converges to

a limit, even if the limit cannot be found explicitly. As mentioned at the end of

section 4.1, if we allowNto tend to infinity, the partial sum

SN=

∑N

n=1

un

of a series may tend to a definite limit (i.e. the sumSof the series), or increase

or decrease without limit, or oscillate finitely or infinitely.

To investigate the convergence of any given series, it is useful to have available

a number of tests and theorems of general applicability. We discuss them below;

some we will merely state, since once they have been stated they become almost

self-evident, but are no less useful for that.

4.3.1 Absolute and conditional convergence

Let us first consider some general points concerning the convergence, or otherwise,

of an infinite series. In general an infinite series

∑

uncan have complex terms,

and in the special case of a real series the terms can be positive or negative. From

any such series, however, we can always construct another series

∑

|un|in which

each term is simply the modulus of the corresponding term in the original series.

Then each term in the new series will be a positive real number.

If the series

∑

|un|converges then

∑

unalso converges, and

∑

unis said to be

absolutely convergent, i.e. the series formed by the absolute values is convergent.

For an absolutely convergent series, the terms may be reordered without affecting

the convergence of the series. However, if

∑

|un|diverges whilst

∑

unconverges

then

∑

unis said to beconditionally convergent. For a conditionally convergent

series, rearranging the order of the terms can affect the behaviour of the sum

and, hence, whether the series converges or diverges. In fact, a theorem due

to Riemann shows that, by a suitable rearrangement, a conditionally convergent

series may be made to converge to any arbitrary limit, or to diverge, or to oscillate

finitely or infinitely! Of course, if the original series

∑

unconsists only of positive

real terms and converges then automatically it is absolutely convergent.