Is the calculation of the series in this video correct? I am watching this video (from MIT OCW) and Prof. Jerison is explaining about series. He is trying to calculate that if some blocks of equal length are kept on top of each other, will the last block be able to cross the first one or not.

He has calculated the series and found it is:
$$C_n=1+1/2+1/3+1/4+\ldots1/n$$

Using this, he showed that he can take the blocks as far as he wants. But according to me, the correct series should be $$C_n=1+1/2+1/4+1/8+\ldots+1/2^n$$ and the series converges to $2$, so he cannot take the blocks farther than the edge of the first block.
So am I right? If I am right, please also tell where was the mistake in the video?
 A: When he writes that the next block, the $N+1$th block, has a center that is 1 more than the center of mass of all the of the blocks before it, $C_N$. We're not saying that the next center of mass if $1$ more to the right, we're saying that we're placing the next block $1$ to the right of the previous center of mass. 
And now we want to come up with the expression for $C_{N+1}$, the new center of mass after adding the next block. The center of mass of objects is a weighted average, therefore, we sum the products of mass and position for each block. But we can group the first $N$ blocks into one "block" with a mass of $N$ (we assume that each block has a mass of $1$ to make things simple).
So our new center of mass, $C_{N+1}$, is the mass of the old blocks, $N$, multiplied by the old center of mass, $C_N$, added to the mass of the new block, $1$, multiplied by the new location which we established as being $1$ more to the right of the old center of mass, $C_N+1$ (Note the 1 is not in the subscript), all divided by the total mass of the new system, $N+1$.
This yields the equation:
$$C_{N+1}=\frac{NC_N+C_N+1}{N+1}$$
Algebraic manipulation can transform that into the relation he ended with:
$$C_{N+1}=C_N+\frac{1}{N+1}$$
