Consider the series sigma (from k=1 to infinity) (-1)^k.(x+k/k^2) for x within [0,1].
I deduce that the series (-1)^k.(x/k^2) is uniformly convergent from using the M-test for convergence, and I assume that the series (-1)^k.(1/k) is not uniformly convergent because it has no x term so isn't a series of functions satisfying the definition of uniform convergence of series. (I know that it is conditionally convergent, however.)
Thus is the original series uniformly convergent?