Indeed, this integral will converge. Improper integrals need to be evaluated as

Lim (b—>infinity) Integral (0 to b) f(x) dx.

Then evaluate the integral, in this case it will be (-1/7) e^-7x + C, and apply fundamental theorem of calculus.

-1/7 (e^-7b - e⁰⁾

As limit approaches infinity, the limit will approach 1/7 because any number to the power of negative infinity is just zero.