I've been told that when it comes to uniform convergence of series, only the tail matters, This seems intuitively obvious, but is there a theorem one can refer to?
Further, if $\sum_{m}^\infty f_n(x)$ converges uniformly towards $f$, does $\sum_{n = 1}^\infty f_n(x)$ converge uniformly towards $f + \sum_{n=1}^{m-1} f_n(x)$?
Yes, there's the same theorem for a sequence and that's where this statement follows from. What's more, this is true even if it converges not uniformly. Just write the n-th partial sum of the two series and you will see why.
The theorem for sequences states that: if you remove/add a finite count of elements from/to a sequence that operation does not change its convergence i.e. if the original one converges (does not converge), then the new one also converges (does not converge).
Consider the series $$ \sum_{n\ge0}a_n,\qquad\sum_{n\ge0}b_n $$ where $a_n=b_n$ for every $n>m$ (for a given $m$).
Define $$ c_n=a_n-b_n $$ Then the series $\sum_{n\ge0}c_n$ is certainly convergent to $c_0+c_1+\dots+c_m$. Since $a_n=b_n+c_n$, you know that if $\sum_{n\ge0}b_n$ is convergent, then $$ \sum_{n\ge0}a_n=\sum_{n\ge0}(b_n+c_n)=\sum_{n\ge0}b_n+\sum_{n\ge0}c_n $$ is convergent as well. What's used is the theorem about the sum of two convergent sequences, which converges to the sum of the limits.
You can easily extend this to uniformly convergent series of functions.
