Summary: Tewodros Amdeberhan
DeVry Institute, Mathematics
630 US Highway One, North Brunswick, NJ 08902
Problem 1540: P
a Show that x
n + x ,1n ,x + 1n has a unique non-zero real root rn.
b Show that rn increases monotonically.
c Evaluate limn!1 rn=n:
Solution: Let fnx := x
n + x ,1n ,x + 1n. Notice that if x = ,t 0, then we have
fn,t = ,1n
+ t + 1n
is either always positive or negative depending on n. Thus it su ces to look for the positive roots
of the fn's.
Claim: 9! rn 0 such that fnx 0 for 0 x rn; fnx 0 for x rn; fnrn = 0 and