Suppose you had a summation $\sum(-1)^na_n$, where $a_n$ is the $n$th digit of $e$ and $a_0=2$.
I know it diverges, but I want to know if its possible to evaluate anyways. Since it is alternating, we can use a number of ways to attempt to evaluate it, an Euler transform, take partial sums and averaging...
Any ways, it comes out to be
$$\sum(-1)^na_n=2-7+1-8+2-8+1-8+2-8+4-5+9-\dots$$
There is no clear pattern to me, and when I attempted a few methods to evaluate it, well, I failed, as in I do not know how to find the $n$th digit of $e$ so easily. (And I don't quite understand how to perform an Euler Transform on this sequence, perhaps they must be monotone increasing?)
More generally, is it possible to take the digits of an irrational number and summate them to a finite value? Like an easier sequence from $\sqrt2$?