1. Consider the sequences defined as follows.(an)
=(12,13,23,14,24,34,15,25,35,45,16,26,36,46,56,17, . . .),(bn)
=(n2(−1)n)= (−1,4,−9,16, . . .).(i)...
1. Consider the sequences defined as follows.(an)
=(12,13,23,14,24,34,15,25,35,45,16,26,36,46,56,17, . . .),(bn)
=(n2(−1)n)= (−1,4,−9,16, . . .).(i) For each sequence, give its lim
sup and its lim inf. Show your reasoning; definitions are not
required.(ii) For each sequence, determine its set of subsequential
limits. Proofs are not required.
1 point)
If the series y(x)=∑n=0∞cnxny(x)=∑n=0∞cnxn is a solution of the
differential equation 1y″−6x2y′+2y=01y″−6x2y′+2y=0, then
cn+2=cn+2= cn−1+cn−1+ cn,n=1,2,…cn,n=1,2,…...
1 point)
If the series y(x)=∑n=0∞cnxny(x)=∑n=0∞cnxn is a solution of the
differential equation 1y″−6x2y′+2y=01y″−6x2y′+2y=0, then
cn+2=cn+2= cn−1+cn−1+ cn,n=1,2,…cn,n=1,2,…
A general solution of the same equation can be written as
y(x)=c0y1(x)+c1y2(x)y(x)=c0y1(x)+c1y2(x), where
y1(x)=1+∑n=2∞anxn,y1(x)=1+∑n=2∞anxn,
y2(x)=x+∑n=2∞bnxn,y2(x)=x+∑n=2∞bnxn,
Calculate
a2=a2= ,
a3=a3= ,
a4=a4= ,
b2=b2= ,
b3=b3= ,
b4=b4= .