No the series does not converge. The given problem is the harmonic series, which diverges to infinity.

Does series converge or diverge?

If you've got a series that's smaller than a convergent benchmark series, then your series must also converge. If the benchmark converges, your series converges; and if the benchmark diverges, your series diverges. And if your series is larger than a divergent benchmark series, then your series must also diverge.

Who proved harmonic series diverges?

The series diverges—a fact first demonstrated by Nicole'd Oresme [1, ca. 1323-1382]. There are a number of proofs that the harmonic series diverges, some of them well-known and elementary.