Start at 5, and then add the terms obtained by dividing the
#SUM OF GEOMETRIC SEQUENCE PROOF SERIES#
This works also for other series for instance, if you In algebraic notation, you do as follows: Therefore, the first term of the series was already X/2 Instead of the first, then you get an answer of X/2 instead of X.
That is, if you start to add up at the second term of your series (that is, you do to the whole series what you would normallyĭistributing the division over the sum, you get To find a repetitive pattern, you divide everything by 2 You want to find its sum X, so that you have Where every term after the first is the previous termĭivided by 2. Is to find a way to have a repeating pattern, and then cancel Series is much more fun than the "formula". The "method" of finding the sum of an infinite geometric
Otherwise the infinite sum does not exist Since the sum does not approach a specific value as n increases, we say that the sum of the infinite geometric series does not exist.Ī + ar + ar 2 = ar 3 +.+ ar n-1 +. Suppose that r = -1 then the infinite series is a - a + a - a +. which again increases without bound, and hence the sum of the infinite geometric series does not exist.Ĭase 4. Suppose that r = 1 then the infinite series is a + a + a + a +. In either situation as n incrreases S n does not approach a specific value so we say that the sum of the infinite geometric series does not exist.Ĭase 3. If r < -1 then as n increases r n oscillates between positive and negative values but again increases in size without bound. If r > 1 then as n increases, r n increases in size without bound. I am not going to give a rigerous proof but rather try to illustrate the possibilities.Ĭase 1. To see what happens when n increases you need to consider four cases. Subtraction of the second equation from the first gives RS n = ar + ar 2 = ar 3 +.+ ar n-1 + ar n The usual technique is to multiply both sides by r to get Assume that a is positive and the sum of the finite geometric series is. You have probably already seen the exprression for the sum of a finite geometric series but I want to develope it just to make sure we are using the same notation. If you could either help me with this or point me in the direction of an informative website that could help me, I'd appreciate it.
#SUM OF GEOMETRIC SEQUENCE PROOF HOW TO#
My math book attempts to explain the concept by giving formulas involving sigma and |r|, but it does not really explain how to go about finding the sum of an infinite geometric series. I ran into a problem when studying how to find the sum of an infinite geometric series.
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