For every $k\in\Bbb Z$, let $f_k$ be a Baire 2 class function on $\Bbb R.$ Assume $\sum_{k\in\Bbb Z} f_k$ is convergent. Define $f:=\sum_{k\in\Bbb Z} f_k$ so $f$ is a function. Moreover, $f$ is a Baire class 2.
Let us recall the well know fact: The finite sum of Baire class 2 on $\Bbb R$ is still Baire class 2. Actually, it is still true for any Baire class $\alpha$, where $\alpha<\mathfrak c.$
Here is the outline of the proof: since each $f_k$ is Baire class 2 so there exists a sequence $\{g_{nk}\}$ of Baire class 1 function that is convergent pointwise to $f_k.$ We need to find a sequence $\{h_n\}$ of Baire class 1 that convergent pointwise to $f.$ Now, Define a sequence $\{h_n\}$ as follows $$h_n=g_{n1}+g_{n2}+\dots+g_{nn}$$ Notice that $h_n$ is Baire class as finite sum of Baire class 1 function as we mentioned above. Also, since $f:=\sum_{k\in\Bbb Z} f_k$ is convergent so we can make the tail of this series very small. I do not know how can I express the tail for this series since it has negative indexes. I think I can finish if I figure out this one. Any idea.