Sum Math Problem: Find Condition for Finite Sum of Positive Real Numbers

[mercedesbenz]

Let $$u_n$$ be a sequence of positive real number.
If $$\sum_{n=1}^{\infty}u_n^{2}$$ finite + (condition??) then $$\sum_{n=1}^{\infty}u_n$$ finite.
I want to find the condition.Please help me.

 

[Defennder]

Are you looking for a necessary or a sufficient condition?

 

[colby2152]

and real?

 

[VietDao29]

Let $$u_n$$ be a sequence of positive real number.
If $$\sum_{n=1}^{\infty}u_n^{2}$$ finite + (condition??) then $$\sum_{n=1}^{\infty}u_n$$ finite.
I want to find the condition.Please help me.

IIRC, then there is a theorem like this:

Given the sequence of positive real number (u_n)

The series $$\sum_{n = 1} ^ {\infty} u_n$$ converge, if and only if $$\lim_{n \rightarrow \infty}(u_n \times n ) = 0$$.

Let's see if you can prove this theorem. :)

Now, using the above theorem, can you try to work out the problem? :)

 

[mercedesbenz]

thank you so much for your advice,VietDao29.I will try to do it again.