## Example 6.1.1:

Consider the function

*f*, where*f(x) = 1*if*x 0*and*f(x) = 2*if*x > 0*.- The sequence
*{1/n}*converges to*0*. What happens to the sequence*{ f( 1/n ) }*? - The sequence
*{ 3 + (-1)*is divergent. What happens to the sequence^{n}}*{ f(3 + (-1)*?^{n})} - The sequence
*{(-1)*converges to zero. What happens to the sequence^{n}/ n }*{ f((-1)*?^{n}/ n ) }

- For the first sequence, we clearly have that
*f(1/n) = 2*for all*n*. Hence, the limit of*f(1/n)*equals*2*. So a function applied to a sequence results in a new sequence that can converge to a different number. - For the second sequence, we know that
*3 + (-1)*for all^{n}> 0*n*. Hence,*f(3 + (-1)*for all^{n}) = 2*n*. This time, a function applied to a divergent sequence results in a convergent one. - For the third sequence we know that alternating terms switch
signs. Hence,
*f((-1)*if^{n}/ n) = 1*n*is odd, and*2*if*n*is even. But then the resulting new sequence is divergent. Hence, we have an example where a function can turn a convergent sequence into a divergent one.