# Thread: The integral test and decreasing functions...

1. ## The integral test and decreasing functions...

hello,

I am studying the integral test at the moment and I understand that in order to be able to use the integral tests on series to find whether they converge or diverge is by making sure that the function is continuous, positive and decreasing.

I am having problem with proving that the function is decreasing...

for example i got the following problem:

$\100dpi&space;\sum_{&space;n&space;=&space;2}^{\infty}&space;\frac{n}{2n^{3}+1}$

I know that I could use something called the first derivative test which requires me to take the derivative of the function and then find the critical points but I am not sure how that can help me prove that the function decreases... Any help would be greatly appreciated.

thank you.

2. For $f(x)=\frac{x}{2x^3+1}$, Clearly its positive and continuous for $n \geq 1$.
Now, we need to show that it is decreasing.
Also, What is question exactly? Is "test the series for convergence" or "use the integral test to ..." ?

3. The derivative equals

$\100dpi&space;\frac{-4x^{3}+1}{(2x^{3}+1)^{2}}$

and the question simply asks to find whether the series converges or diverges.

thanks.

4. Originally Posted by dmitrip
The derivative equals

$\100dpi&space;\frac{-4x^{3}+1}{(2x^{3}+1)^{2}}$

and the question simply asks to find whether the series converges or diverges.

thanks.
Well, Clearly $-4x^3+1=1-4x^3 < 0 \,\ \forall x \geq 1$
Hence, the numerator is negative, and clearly the denominator is positive, It follows that $f'<0$ for $x \geq 1$.
And this proves the function is decreasing.
Now, focus on the convergence/divergence of $\int_1^{\infty} \frac{x}{2x^3+1}dx$.