Thread: Question about series

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 ??

Intermediate value theorem gives you that exists a zero, but not the fact there is exactly one. But $\displaystyle f_n$ is increasing in $\displaystyle \left(\frac 1{n+1},\frac 1n\right)$.

Since fn increasing on the interval , so $\displaystyle x_{n+1}-x_{n} > 0 $

right ??
but how we can find $\displaystyle a_{n} $

I am a little confused
but maybe because of the function is bounded and it has a limit . so it has a one zero point