Question

A series question: Taylor series allow us to write a f(x) in term of series.But what if given series and to solve for f(x)?
For example,if I have 1+1/2+1/4+1/8+1/16+...1/2`n... (n goes to infinite) then how can I combine it with 1/(1-x)?Can I solve all the power series using something like inverse Taylor series?? I can not make sure a function just by knowing f(o)=1,f`(0)=1,f``(0)=2...

Answer 1

With the given values of a function's derivatives we can find the function, this is what differential equations are, but for infinite derivatives it is a tricky task.
It is better to solve series by observation and trying to fix what an arbitrary term looks like. For example, here you can see that each successive term is a further multiple of 1/2, also the first term is 1, so the n term would be of the form\[a_n = \frac{1}{2^{n-1}}\]Put this into summation for the range n = 1 to n = infinity and you'd get the answer.

Answer 2

thats correct..!!!