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 ??