Princeton Review Typo Page 77
Posted: Sat Aug 22, 2009 9:50 am
3rd edition. Solution to example 2.27
"An = f(n) then sequence (An) converges to L <=> f(x) converges to L as x -> infinity"
I think the statement is wrong in that (An) converges to L do not => f(x) converges to L as x -> infinity.
Suppose the statement is true.
Counter example :
Let f(x) be a function s.t. f(x) = M for all x that are no natural numbers. Where M not equal L. And f(x) = Ax for x that are natural numbers. So this function satisfies An = f(n) for all natural numbers n but has the value M for all inputs that are non-natural numbers.
Although (An) converges to L, this function f do not as x -> infinity.
"An = f(n) then sequence (An) converges to L <=> f(x) converges to L as x -> infinity"
I think the statement is wrong in that (An) converges to L do not => f(x) converges to L as x -> infinity.
Suppose the statement is true.
Counter example :
Let f(x) be a function s.t. f(x) = M for all x that are no natural numbers. Where M not equal L. And f(x) = Ax for x that are natural numbers. So this function satisfies An = f(n) for all natural numbers n but has the value M for all inputs that are non-natural numbers.
Although (An) converges to L, this function f do not as x -> infinity.