Show that 1+1/2+1/3+1/4+...is infinite.

Let :

S=1+(1/2)+(1/3+1/4)+(1/5+1/6+1/7+1/8)+(1/9+...

Now, let us consider the series T given by:

T=1+(1/2)+(1/4+1/4)+(1/8+1/8+1/8+1/8)+(1/16+...

Now, 1=1, 1/3 + 1/4< 1/4 + 1/4, 1/5+1/6+1/7+1/8 > 1/8+1/8+1/8+1/8.

And, accordingly:

T<S for any finite number of terms.

T=1+1/2+1/2+1/2+... which is infinite for infinitely many terms.

Consequently, it follows that S must also be infinite.