# Thread: Convergent Series and Orthonormal Sets

1. ## Convergent Series and Orthonormal Sets

What is an example of a convergent series $\sum e_{k}$ that does not have a sum of x?

2. ## Re: Convergent Series and Orthonormal Sets

If you just chose thing pretty much at random, you would probably get such an example. It is only as long as the vectors $e_k$ form an orthonormal basis for whatever vector space you are talking about that this sum will be equal to x.

An example is $e_1= <2, 0>$, $e_2= <0, 3>$, and $x= <1, 1>$.