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Convergence on an Infinite Series - Using Algebra

   I have also seen other examples wherein an infinite series is converged to a finite number.

   1/2 +1/4 + 1/8 + 1/16 + 1/32 + … = 1

   The ratio between each term is 1/2, in other words 1/2 of 1/2 is 1/4, and 1/2 of 1/4 is 1/8, and 1/2 of 1/8 is 1/16, and 1/2 of 1/16 is 1/32, ad infinitum.
   First, they name the entire infinite sum "S;" therefore, S = 1/2 +1/4 + 1/8 + 1/16 + 1/32 + …
   Next, they multiply S by 1/2 (the ratio): S/2 = 1/4 + 1/8 + 1/16 +1/32 …
   Then they subtract: S – S/2 = 1/2
   Then they simplify: S/2 = 1/2
    And finally: S = 1

    In actuality, since S is an infinite sum, it can come very close, but it will never reach 1. .
   Also, mathematicians consider it a fact that to multiple infinity by 1/2, the answer is still infinity.

Converging Absolutely

   This is another concept that needs to be discussed in order to clear up any misunderstanding. When mathematicians write about a series—any series, including a geometric series, and they say it is converging absolutely, they are not stating that the series is completely transforming from an infinite series to a finite number. They are not saying that now the answer actually is "2" with no infinite component. Instead they are indicating that a series of absolute values (all the plus values of a series) are converging on a finite number. But, it is still understood that the absolute values are infinite in length and that they will never reach that finite number.

Why Do Mathematicians Converge an Infinite Series and Give it a Finite Number?

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