Does there exist an infinite convergent series s for any given irrational number a such that the sum of the series s is a?
In other words: Is every irrational number the sum of an infinite convergent series?
Further: Given an irrational number a, is there a way beyond trial and error to find one or more infinite series that converge to a?
I'm compiling the answers in the comments.
If you do not require the numbers in the series to be rational, the series: $$(a,0,0,0,0,\dots) \quad\text{and} \quad \left(\frac a2, \frac a4, \frac a8,\frac a {16},\dots\right)$$
Have sums converging to $a$.
If instead you require them to be rational, the decimal expansion is a good starting point:
$$(3,0.1,0.04,0.001,0.0005, \dots) \quad\text{and}\quad (4, -0.8, -0.05, -0.008, -0.0004, \dots)$$
can both have sums converging to $\pi$, for example.
For a more interesting example, see Greedy algorithm for Egyptian fractions: $$(3, \frac18,\frac1{61},\frac1{5020},\dots)$$
So the answer to your question is affirmative, if the numbers of the series are taken from $\mathbb R$ or $\mathbb Q$. This is the basis for $\overline{\mathbb Q} =\mathbb R$, after all.