Ignoring the 1, if you group together the first term, then the next two terms, then the next four terms, and so on, you get:
$$(1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + \cdots$$
which is greater than
$$(1/2) + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + \cdots$$
where now each group is exactly equal to 1/2. This shows that the sum of the first $2^n$ terms is at least $1 + 1/2 \cdot n$, and so the sum of all the terms is unbounded.
If you know a little bit of calculus, $\int dx/x=\log x$, so $1+1/2+1/3+\cdots+1/n\ge\int_1^ndx/x=\log n-\log1$.
See Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$ which discusses precisely how fast the series diverges.