Is the sequence of sums of inverse of natural numbers bounded?

This is a classic example of how our intuition can be wrong concerning infinities. As @Kavi said in the comments, this is the harmonic series and indeed diverges. I believe that Spivak provided this as an example to “chew on” in order to show how non trivial boundedness can be, as on first glance many people believe this sequence should be bounded.

Indeed, you can notice that

$$a_1=1,a_4>2(=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}), a_8>\frac{5}{2}(=a_4+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}),a_{16}>3(=a_{8}+\frac{1}{16}+\frac{1}{16}+..8times)$$, so the general pattern is $a_{2^{2n}}>n+1$, so given an upper bound M, you can find a natural number n s.t. $n\geq M$ by archimedian property and so $a_{2^{2n}}>n+1>M$ and hence it can't be bounded.