What is an example of a convergent series $\displaystyle \sum <x, e_{k}>e_{k}$ that does not have a sum of x?
If you just chose thing pretty much at random, you would probably get such an example. It is only as long as the vectors $\displaystyle e_k$ form an orthonormal basis for whatever vector space you are talking about that this sum will be equal to x.
An example is $\displaystyle e_1= <2, 0>$, $\displaystyle e_2= <0, 3>$, and $\displaystyle x= <1, 1>$.