fn(x)=1/(x+1/(2n)) and fn(x)=1/(x+1/(n^2)).

Question

Q1 prove piecewise convergent on (0,infinity)

Q2 prove uniform convergent on [1,infinity)

 

Please help with these two equations. Much Thanks, it's preparation for final exam

 

Answer 1

Let ε>0, x>0. Choose N=1/(2εx2) (same N works for the second problem). Then for any n>N :

1. |fn(x)-f(x)|=1/(2nx(x+1/(2n))<1/(2nx2)<ε. For uniform convergence on [1, \infty) choose N=1/(2ε) to get

|fn(x)-f(x)|=1/(2nx(x+1/(2n))<1/(2nx^2)<1/(2n)<ε.

2. |fn(x)-f(x)|=1/(n^2x(x+1/(n^2))<1/(2n^2x^2)<1/(2nx^2)<ε. Again, for the uniform convergence choose N=1/(2ε) to get

|fn(x)-f(x)|=1/(n^2x(x+1/(n2))<1/(2n^2x^2)<1/(2nx^2)<1/(2n)<ε.

QED

Answer 2

Let ε>0, x>0. Choose N=1/(2εx²) (same N works for the second problem). Then for any n>N :

 

1. |f_n(x)-f(x)|=1/(2nx(x+1/(2n))<1/(2nx²)<ε. For uniform convergence on [1, \infty) choose N=1/(2ε) to get

 

|fn(x)-f(x)|=1/(2nx(x+1/(2n))<1/(2nx^2)<1/(2n)<ε.

 

2. |fn(x)-f(x)|=1/(n^2x(x+1/(n^2))<1/(2n^2x^2)<1/(2nx^2)<ε. Again, for the uniform convergence choose N=1/(2ε) to get

 

|fn(x)-f(x)|=1/(n^2x(x+1/(n2))<1/(2n^2x^2)<1/(2nx^2)<1/(2n)<ε.

 

QED