infinite series (idea) by salve las ballenas

The proof of the formula for the sum of infinite terms of a geometric series, S=a/(1-r), is quite simple.

First, recall the formula for the sum of finite terms of a geometric series, S_n=(a(1-rⁿ))/(1-r)

Replace n with infinity.

Again, recall that r's value can be defined as -1>r>1, and that as fractions are multiplied, their product gets closer to 0. SO, r^infinity is essentially equal to 0, and 1-r^infinity is essentially equal to 1, making a(1-r^infinity) equal to a

Thus, as the term n is no longer needed in the formula (as the quanitiy (1-r^infinity) is equal to 1), niether is n needed in the sum, yielding the formula S=a/(1-r).

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