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.)  

           
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