This is because the partial sums of the series have logarithmic growth.

In order to investigate this more carefully, the partial sums need to be defined:

However, this optimal partial sum will usually have more terms as the argument approaches the limit value.

A series is convergent if the sequence of its partial sums converges.

In other words, we construct partial sums of the b terms:

The last partial sum is the value for b.

The first column then contains the partial sums of the Euler transform.

He does not quite take the limit of the above partial sums, but in modern calculus this step is easy enough:

Similarly, the sequence of even partial sum converges too.

This understanding leads immediately to an error bound of partial sums, shown below.