For the purpose of understanding convergence,
consider the following quotes by the mathematics professor, Dr. Lee
Lady, from the University of Hawaii.
"Some Series Converge: The Ruler Series
"At first, it doesn't seem that it would ever
make any sense to add up an infinite number of things. It seems that
anytime one tried to do this, the answer would always be infinitely
large. The easiest example that shows that this need not be true is
the series I like to call the 'Ruler Series:'
"1 +1/2 +1/4 + 1/8 + 1/16 + 1/32 + …
"We say that the series, 1 +1/2 +1/4 + 1/8 + 1/16 +
1/32 + … , converges to 2.
"Symbolically, we indicate this by writing: 1 +1/2 +1/4
+ 1/8 + 1/16 + 1/32 + … = 2
"This notation doesn't make any sense if interpreted literally,
but it is common for students (and even many teachers) to interpret
this as meaning, 'If one could add all the infinitely many terms,
then the final sum would be 2.' This, unfortunately, is not too much
different from saying, 'If horses could fly, then riders could chase
clouds.' The fact is that horses cannot fly and one cannot add
together an infinite number of things. Instead, one is taking the
limit as one adds more and more and more of the terms in the series.
"The fact that one is taking a limit rather than adding an
infinite number of things may seem like a fine point that only
mathematicians would be concerned with. However certain things
happen within infinite series that will seem bizarre unless you
remember that one is not actually adding together all the terms."^{4
}
Note: All the numbers cannot be added because the numbers are
infinite. And, upon further inspection, it will be realized that the
'limit' is a finite number that is converged upon, but never reached.
(In divergence, the series continues forever and never comes to a limit
or a finite number, to converge upon.)
