178 13 Power Series and Special Functions(a) We have
u U\ U2 U"1 u-2 «nn w nil
n = "o^1 U Vra = 0,1,2,...
0 U\ Un_!< q < 1 =>• un < u 0 qn, q < l,Vn.and therefore
Mfc + l
Uk
Since the series ]T^ «o<Zn is convergent, the series ^^° Uk is convergent.
Mfc+l
Uk> 1 => M„ > wo, Vn.Since the series «o + «o + • • • is divergent, so is 53^° Uk-
(b) linife^oo ^=?<1^V£>03? + £<1;TO have 2±±i < q +
e < 1, k > N, where iV is sufficiently large. Then, the series Y^N uk 's
convergent and hence so is Y^f wfc. On the other hand,lim ^±l=q>l^^±l>1^k>N
fc-+oo Uk Uk
for sufficiently large N, and therefore 53 o° u^ is divergent. •Theorem 13.1.10 (Cauchy's Test) Let 53o° Uk be a series with positive
terms,
(a)
oo
(uk)* < q < 1,VA; =£• the series 2_]uk *s convergent.
o
oo
(uk)J > 1,Vfc =>• £/ie series ]>Uk is divergent.
o
(b) If'limfc_>oo(wfc)E = 9; £/&en *^e series £Zo°ufc «s convergent for q < 1 and
divergent for q > 1.Remark 13.1.11 Lei a series be convergent to a sum S. Then, the series
obtained from this series by rearranging and renumbering its terms in an ar-
bitrary way is also convergent and has the same sum S.
13.2 Sequence of Functions
Definition 13.2.1 A sequence of functions (/„}, n = 1,2,3,... converges
uniformly on E to a function f if
Ve > 0, 3N e N 3 n > N =» \fn(x) - f{x)\ < e, Vx e E.Similarly, we say that the series Y fn(x) converges uniformly on E if the
sequence (Sn) of partial sums converges uniformly on E.