# Thread: Function series which converges uniformly but not absolutely

1. ## Function series which converges uniformly but not absolutely

Hello,

I'm new here and would like some help finding a function series qualifying the conditions in the title.

Thanks!

2. Take $\displaystyle f_n\left(x\right) =\frac{\left(-1\right)^n}n$ for $\displaystyle x$, for example, in $\displaystyle \left[0,1\right]$. The function is constant so the convergence is uniform and it doesn't converge absolutely.

3. Thanks, but when I say series I mean

4. Originally Posted by Mosho
Thanks, but when I say series I mean
That's what he gave you. Let $\displaystyle g(x)=\sum_{n=1}^{\infty}\frac{(-1)^n}{n}$. This converges uniformly to $\displaystyle \ln\left(\frac{1}{2}\right)$ for a number of reasons...the easies to just state is Abel's Uniform Convergence Test -- from Wolfram MathWorld with $\displaystyle a_n=\frac{(-1)^n}{n},f_n(x)=1$.

5. My bad then. I had an exam today and what you guys said is what I wrote, but most people said I was wrong.

Thanks alot

6. I have another question:

Is this statement true/false?

"The sum of a absolutely convergent series and a semi convergent series is a semi convergent series"

Thanks again!

7. Originally Posted by Mosho
I have another question:

Is this statement true/false?

"The sum of a absolutely convergent series and a semi convergent series is a semi convergent series"

Thanks again!
What does semi-convergent mean?

8. Sorry, conditionally convergent.