I will start typing it off line and I should be able to post it later other things being equal.
relpace the sum by an integral which is an upper bound for the sum.
ignore the first two terms outside the summation as they are always negative
show that for k>13 the integral and the last two terms outside the summation are strictly decreasing as a function of k, and that at k=14 this is less that 1.
This shows that the function is bounded above by 1 for k>13, and so we need only check the first 13 terms to prove that f(k) is less than 1 for all positive k.
We also seem to have a discrepancy with our evaluation of the first few valuea of f(k), maybe these need checking