(1) is the limit of the partial sums!
That is, (1) equals
lim_{m --> infinity} SUM {from n = 1 to m} 9/10^n.
The limit of this indeed equals 1 (since you're in
a "serious" maths degree, I'll leave the proof to
you).
Again, this is not the limit of the sequence a_n = 9/10^n, it is the limit of the partial sums.
Now, how is it that 0.9999... is NOT equal to 1?
It HAS to be, because 0.9999... is EXACTLY the
infinite sum of the above. That is what
0.999999... means.
It's kind of funny that blizzard posted this
as an april fools joke thinking it wasn't true,
but it is in fact true.
Notice it cites:
R.V. Churchill and J.W. Brown. Complex Variables and Applications. 0.9999... = 1 ed., McGraw-Hill, 1990.
and
W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.
Rudin is a serious mathematician
and he knows what he's talking about--
not that an argument from authority means
anything:)
Hence, in conclusion, 1 = 0.99999....
Slashdot Mirror
(1) is the limit of the partial sums! That is, (1) equals
:)
Hence, in conclusion, 1 = 0.99999....
lim_{m --> infinity} SUM {from n = 1 to m} 9/10^n.
The limit of this indeed equals 1 (since you're in a "serious" maths degree, I'll leave the proof to you).
Again, this is not the limit of the sequence a_n = 9/10^n, it is the limit of the partial sums. Now, how is it that 0.9999... is NOT equal to 1? It HAS to be, because 0.9999... is EXACTLY the infinite sum of the above. That is what 0.999999... means.
It's kind of funny that blizzard posted this as an april fools joke thinking it wasn't true, but it is in fact true.
There was (is) a rather large discussion of this on sci.math. Here's a sample link: http://mathforum.org/dr.math/faq/faq.0.9999.html
Notice it cites:
R.V. Churchill and J.W. Brown. Complex Variables and Applications. 0.9999... = 1 ed., McGraw-Hill, 1990.
and
W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.
Rudin is a serious mathematician and he knows what he's talking about-- not that an argument from authority means anything