Series

A different application of non-convergent series is the rep resentation of functions by means of so-called asymptotic series. These series when carried to infinity are not convergent but are such that it is possible to estimate at any stage the order of the difference between the function and the expansion, and it may happen that a few terms of the expansion give a good approxima tion. For a function of a variable x, such expansions are usually power series in i/x, and the difference between the value of the function and the first n terms of the expansion is 0 (n+ 1) . Now (n+ i)!xn+' decreases so long as n+ i is less than x, though afterwards it increases without limit, and when x is very large the least value of (n+ I) is very small, so that the first n terms of the expansion will give a very good approximation to the value of the function. One of the best known of such ex pansions is Stirling's formula, log (n!) = (n+ Vogn — n+ I log ( 27r) +9/12n, where o <0 < I. The term 0/12n can be replaced by a formal development as a power series in but it is not convergent for any value of n.

Summation of Series. Asymptotic Representation of the Sum of n Terms of a Series.—If u„ can be decomposed into a number of expressions of the form a"cp(n), where a is some fixed number, real or complex, and 4)(n) is a polynomial in n, then it is possible to find a compact expression for s,,, but there is no other type of series for which the same property holds with complete generality. Included in this type are the ordinary arithmetic and geometric progressions; for the harmonic series and generally no exact expression for s„ can be found. The sum of the first n terms of the series Znr, where r is a positive integer, can be expressed as a polynomial in n or n+1, of degree r+ I, by means of the identities is the expansion of x/ (ex — I) as a power series in x.

The numbers B3 • • ' are called Bernoulli numbers (q.v.). They have been tabulated as far as B90. The second polynothial involves the coefficients in the expansion of x/ that is, x { (0' — — (ex — , which can therefore be expressed in terms of Bernoulli numbers. If r is odd, En' can be expressed as a poly nomial AO of degree Z (r+ I) in which denotes and (2n+ i)f()/r. Unless r is very large the coefficients in the polynomial can be determined fairly easily from the identity le The sum of the rth powers of the first n odd integers can be expressed as a polynomial of degree r+ I in n, containing no odd powers of n, if r is odd, and no even powers of n, if r is even, divisible by n in either case.

It is often possible to obtain asymptotic formulae for the sum of the first n terms of a series. Two functions 4)(n) and tk(n) are said to represent each other asymptotically if the ratio 4)(n) / tk (n) tends to unity as n tends to infinity and we write If u„ = 4)(n), which is steadily decreasing, tends to a finite limit and we can write Thus, if Stirling's formula for log(n!) is an asymptotic formula for s„, when The limit of i+1+ • • • is an important number in analysis. It is called Euler's constant

and is denoted by C or sometimes by 7.

It is often possible to find the sum of a convergent series although no expression for s„ can be obtained. By means of the infinite products for the trigonometrical functions it can be shown that the sum of when r is an even positive integer, is a ration al multiple of 7rr, and that the sum of I( — i)"(2n+ i) r, when r is an odd positive integer, is a rational multiple of irr. The sum of a series can be found occasionally by the help of Euler's constant. Thus the sum of the first 2n terms of the series Z( is equal to —s„, where which is equal to —log 2n) — loge) +log 2.

Since —log 2n) and s„ —loge each tend to C, Euler's constant, the sum of the series is log 2. For since the series is convergent its sum is the limit of the sum of the first 2n terms. By a some what similar method we can prove that the sum of the series /( — i)n(logn)/n is C log 2 — Elog Methods of obtaining approximations to the sum of a series and of converting slowly converging series into more rapidly converging series belong to the Calculus of Finite Differences (q.v.).

Infinite Products.

If from a sequence of numbers ) we form a sequence in which p„= v, v3 • then, if Pn tends to a finite limit, distinct from zero, the infinite product is said to be convergent. The infinite product is said to diverge to infinity or to zero, if p,„ tends to infinity or to zero. If p„ oscillates, the infinite product is said to oscillate.

If the product is convergent must tend to unity, so that must be of the form I where N. tends to zero. In considering the question of the convergence of infinite products we may, without any loss of real generality, consider only products in which lu„1 < I for all values of n.

The convergence of an infinite product can be discussed directly by means of certain inequalities, due to Weierstrass (see Brom wich, Theory of Infinite Series); but the results may be obtained readily from the fact that the infinite product 11(i is con vergent, if the series Elog(r -Fu„) is convergent, divergent to +00 (or to zero), if the series is divergent to +00 (or — 00 ), and oscil latory, if the series is oscillatory. If is complex, log(' is the principal value of the logarithm: that is, the value which tends to zero with u„. We see from the above why we regard an infinite product as diverging to zero rather than converging to zero.