Thread: Proof Question sequences

Hello.

I am having trouble proving this problem.

Let (xn) be a sequence of positive numbers such that lim [(xn)^(1/n)] exists and equals a. a is greater than 0, but less than 1. Prove that lim (xn) = 0.

The limits are to inf.

Thank you.
Timothy.

If 1>r>0 then the sequence r^(1/n) converges to 1.
So if glb(x_n)>0 that would contradict the given that a<1.
Now use the fact the glb(x_n)=0 to prove the statement.

Originally Posted by THulchenko

Hello.

I am having trouble proving this problem.

Let (xn) be a sequence of positive numbers such that lim [(xn)^(1/n)] exists and equals a. a is greater than 0, but less than 1. Prove that lim (xn) = 0.

The limits are to inf.

Thank you.
Timothy.

If
0<lim[(xn)^(1/n)]<1
Then by the root test,
The sequence of partial sums converges,
SUM xn -----> Converges.
That means the sequence,
{xn}----> 0
Because the sequence of the series always converges to zero whenver the infinite series exists.