Math Help - Question about series

Show that this function
has exactly real zero point and it is in the interval


decide if the series
converges absolutly or conditionally ??
For which x converge the power series

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

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 is increasing in .

Since fn increasing on the interval , so

right ??
but how we can find

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