Sum simplification



I have the following sum, which is coth(x)-1/x:
sum(((4^i * bern(2 * i) * x^(2 * i - 1))/((2 * i)!)),i,0,inf) - (1/x).
Is there a maxima or a commercial macsyma function
which will evaluate the first term of the sum, perform the
simplification, and return a sum starting from i=1?
Or do I have to transform this by picking apart the expression by hand.

Thanks,
Dan Stanger