12.4 Power series 629and it is worthwhile to be able to start with a clean sheet of paper and write all
of the formulas needed to derive them.
21 If lanlltn < RI for each sufficiently great n,prove that Eax" converges
when ixi < 1/RI. Solution: When lanli"n < R1, we find that
lanll1nlxl < IxRil, Ia"x"I < IxR,I".The hypothesis that ix! < 1/R, implies that ixR,i < 1, so ZixRii" < oo and con-
vergence of Eanxn follows from the comparison test.
22 If R2 > 0 and lanilt" k R2 for an infinite set of values of n, prove that
Ea"x" diverges when jxI > 1/R2. Solution: When Ianll/" >_ R2, we find that
lanil/nixl = R2IxI, Ianx"i (R2ixi)".The hypothesis that jxi > 1/R2 implies that R2ixI > 1. Hence la"x"i >_ 1 for
an infinite set of values of n. Thus it cannot be true that
lim anx" = 0,n--
and therefore Xanx" must be divergent.
(^23) Prove the following famous theorem, which is known as the Abel power
series theorem. If the series in
(1) s=ao+a,.+a2+aa+...
converges to s, then the series in
(2) f(r) = ao + air + a2r2 + aara +
converges when 0 < r < 1 and
(3) lim f(r) = s.
r-41-
Solution: Let s, = ao + a, + + an, so that
(4) lim S. = s.
n-. w
There must be a constant M such that Iaki <- M and Iskl < M for each k. When
Iri < 1, the series in (2) and the series in
(5) f(r) = so + (Si - so)r + (s2 - si)r2 + (S3 - S2)ra +
are therefore both convergent and they both converge to the same numberf(r)
because so = ao, s, - so = ai, s2 - s, = al,. Because the separate series
are both convergent, it follows from (5) that
(6) f(r) (So + SIT + S2r2 + .. -) - (0 + sor + sir' +. .. .)
and hence that
(7) f(T) _ (1 - T) (SO + SIT + .c2T2 +. .).
But
(8) s = (1 - r)(s + Sr + sr2 + ...)
and hence
(9) f(T) - s = (1 - r) L.l (sk - s)T.
k-0