Question

Q7 You are given the sequence n2 In(x)= n2 + of functions on the domain [0, oo) where 0 < a < 2. Determine the range of o for which the sequence is uniformly convergent.

Answer 1

fo(x)= no in [0,00) and o<o<2 ne+g3 We know that A Junction of Sequence focx) in Uniformly convergent iff Mnto as n700 e mn= Sup I fora)- fro) where D= domain XED lim foco) = Din naar 120 n2tag [co] Joom By L Hospital Rule lim anarx? 2 2 - Dim no ano is n o fila) >0. Here ad o<a<2 fro= lim t fra) = 0 lip filx) Now Mn= Sup Inal I n2403- XE[0,oo Now we find the to point in fila) give the Superemum. coooo) when Inial nga n2+x3 deoniuntive with reapect JAG ) to' Ins) = (n2+ 3) (anda) - 1092) (372) (n+73)2 and that anday 3n49=0 20942 go-nag=0

nla [an2_23]=0 > x=0 dB and * = (2n233) at X= (202) where Inra) give the Supermum in corso Pot x = (202) in ean we get Mo= n41202% n2+ ane 302 sen no 3 lim Mn = lim g a + 2/3 QtY3-2 n> 2132-33 =lim 23 n9+976 :23 na no 3 We know that food is U. Convergent i fyl Mnyo aa n oo. In above Question is ax then Mnoo as n o ip Inca) is a not U. convergent. Case-I 2<2/3 then Mn 0 ag nyoo 7 fila) is u. when & € [012). ie the Range of af 10, 23) when fira) is uniformly convergent,
in above question we know that Mn test a sequence fn(x) is u.c iff Mn tend to zero as n tend to infinity .