Principles of Mathematics in Operations Research

176 13 Power Series and Special Functions

Proposition 13.1.2 (Cauchy's criterion) The series

oo

k-0

is convergent if and only if

Ve > 0, SN 3 \fn,p e N,n > N, \un+1 + ••• + un+p\ = \Sn+p - Sn\ < e.

Remark 13.1.3 In particular, putting p — 1 we see that ifJ2T=oUk *s con~

vergent its general term Uk tends to zero. This condition is necessary but not

sufficient!

Definition 13.1.4 The series are called the remainder series of the series

E

oo

fe=0Wfc;

oo

U„+l + W„+2 H = 2_, un+k-

fc = l

Since the conditions of Cauchy 's criterion are the same for the series and its

remainder series, they are simultaneously convergent or divergent. If they are

convergent, the remainder series is

m

n—+oo *lim S~] u—* n—>oo n+k = lim (S„+m - S„) = S - S„.

fc=l

// the series are real and nonnegative, its partial sums form a nondecreas-

ing sequence Si < S2 < S3 < • • • and if this sequence is bounded (i. e.

Sn < M, n = 1,2,...), then the series is convergent and its sum satisfies

the inequality

lim Sn = S < M.

n—>oo

// this sequence is unbounded the series is divergent linin-^oo Sn = 00. In this

case, we write J^J*Lo uk — 00 and say that the series with nonnegative terms

is divergent to 00 or properly divergent.

Example 13.1.5 The nth partial sum of the series 1 + z + z^2 + • • • is

S„(z)= i_z forz^l.

If \z\ < 1 then

If\z\>l then

zn+l = \z\n+ -> 0, that is zn+1 -> 0 as n -> 00.
-> 00.
Finally, if \z\ = 1 then zn+1 = cos(n + 1)9 + isin(n + 1)6, where 6 is the
argument of z, and we see that the variable zn+1 has no limit as n —> 00
because its real or imaginary part (or both) has no limit as n —• 00. For
z — 1, the divergence of the series is quite obvious.
We see that the series is convergent and has a sum equal to (1 — z)"^1 in
the open circle \z\ < 1 of the complex plane and is divergent all other points
z.