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Luke77
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Just wondering, is there a way to sort of "collapse" a finite series (to get the sum) that isn't classified as arithmetic, geometric or a p-series.
Luke77 said:Just wondering, is there a way to sort of "collapse" a finite series (to get the sum) that isn't classified as arithmetic, geometric or a p-series.
HallsofIvy said:Since you use the word "collapse", there is always the "collapsing" or "telescoping series". For example, it is easy to show that the series [tex]\sum_{i=1}^n \frac{1}{n^2+ n}[/tex] sums to [itex]1- 1/(n+1)[/itex]. That is because, using "partial fractions", we can rewrite [tex]\frac{1}{n^2+ n}= \frac{1}{n}- \frac{1}{n+1}[/tex] so that each "1/k" term reappears as "-1/(k+1)" and cancels. The only terms that survive are the first, 1, and the last, -1/(n+1).
Luke77 said:I'm solving a series with just one coefficient, n, and an exponent. I'm aware that I can bring the coefficient "through" the integral and solve from there but I don't know what sort of formula to use.
I also know I can just write it out.
Luke77 said:I'm solving a series with just one coefficient, n, and an exponent. I'm aware that I can bring the coefficient "through" the integral and solve from there but I don't know what sort of formula to use.
I also know I can just write it out.
The formula for finding the sum of a finite series is: S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term.
If the terms of a finite series are not in a simple arithmetic pattern, you can use the formula: S = (n/2)(a + a(r^n-1)), where r is the common ratio between terms.
No, the sum of an infinite series cannot be found as it has no ending point. However, we can approximate the sum by calculating the sum of a large number of terms.
The sum of a finite series is related to the area under a curve in the sense that the sum can be represented graphically as the area under a curve. This is known as the integral of a function.
The sum of a finite series is significant in mathematics as it helps us to understand the behavior of a sequence or a function. It also has many real-life applications, such as in financial calculations, physics, and computer science.