=5<
series or product is a syrnetric functicn of its terms, or factors,
as the case may be. For, an absclutely convergent infinite series
or product remairs unchanged ir value when its terms, or factcrs,
are rearranged in order. Conversely, every symmetric functicn of
infinitely many elements must be absolutely convergent if it is ex-
pressed as an infinite series, or product, of its elements. For its
valve must remain unchanged by every permutaticn of the rows cf
elements. A conditionally convergent series er product may be made
to have any arbitrary value by rearranging the terms,® or the fac-
tors; * so that a conditicnally cervergent series cr preduct could
not te symmetric. Therefcre, sbsclute ccnvergence and syrretry are
equivalent cerditions for infinite series or preducts.

We shall now preve that if.s,, so1, Soo1, *"**‘, are sbsce

lutely cenvergent, then all of the sis and all of the ¢'s are.

Write S1 = a3 4 Gg + 0s.t Qg 4 -t
$3. T a3 4 69 + ag + a4 + v
Multiplying term by term, S2 = ay°4 az ™ ae®4 a %4 ceeens

Sirce each series called 'sy is aktsclutely ccnvergent, then sp is
absclutely canergent; foriéacb terr a4 cf an absclutely cervergent
series is wultiplied by a factor g¢¢ such that |q¢] < K, a ccnstant,
fcr every i, the resultirg series is absclutely convergent.” Inp
this case ¢¢ is a4, and since s, ccnverges, Loy = C and therefere
satisfies the conditicn that |ay| < K. Using 'sg in the second lire
in'place cf si, we obtain ss in the same ranner. Fcr the sage
reason, 'sy is absclutely ccnvergent. By contiruirg the preocess, we
cbteir s,, s.;'s,, **t*, and ir the same manner sqy, Soe2, Sop, *'*’,
Sco1, ‘Soo2, Soes, *°'*', etc., all of which are atsclutely corver—
gent.

—_——

. Browwich, sec.o8. Smeil, sec. 141.
Bromwich, sec.4i.
Bromwich, sec,318. Smail, sec. 12€.