How and why the name Peking changed to Beijing is a source of sleepless
nights to many modern, western thinkers. Actually, it didn't change. It is
just "spelled" differently. I say "spelled" because Chinese isn't actually
alphabetized, but is phonetically rendered into "English" script in a way
that represents its sounds.
Okay, if you followed that path (pronounced "dao", by the way, and not
"tao") then you can explore the transition from Peking to Beijing. First we
need to understand that our early exposure to anything Chinese came to us
via western scholars, usually priests or missionaries, who were
predominantly English, French, or German. Unfortunately, the same roman
letter in these three languages may represent different sounds.
First, let's deal with the initial sound of the capital of China, the B or
P. Linguistically, "p" and "b" are the same sounds. "B" is the voiced sound.
"P" is aspirated, and a breath takes the place of the humming vocal chords
of the "b." To show that these are the same sound, the linguist designates
them using the same letter, "p". "P" with an apostrophe (p') equals the
aspirated p, and without the voiced p (i.e. the sound of "b" in English).
Now that the initial sound of Peking can be connected to the "Beijing"
pronunciation, we have to deal with the "k" and "j" sound variant, but,
perhaps, this is best left as inexplicable by the fact that it was the
German romanization system that produced "Peking".