Show that this function $\displaystyle f_{n}(x)= x^{5}+nx-1$

has exactly real zero point and it is in the interval

$\displaystyle \left(\frac{1}{n+1},\frac{1}{n}\right)$

decide if the series $\displaystyle \sum \left(-1\right)^{n-1} a_{n}$

converges absolutly or conditionally ??

For which x converge the power series $\displaystyle \sum a_{n}x^{n}?$

I tried to substitute the two end points of the interval for the x in the function by (intermediate value theorm)

to show that we have exactly one zero point , is it useful to use this way ?

After that i tried to take the $\displaystyle \sum x_{n+1}-x_{n}$

Is it useful to solve the problem ?? I got a very complex function , i do not know if it is right to use intermediate value theorm .

Does anyone have an idea ??