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anemone
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Find the exact value of the series \(\displaystyle \frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+\cdots\cdots\)
anemone said:Find the exact value of the series \(\displaystyle \frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+\cdots\cdots\)
The Sum of Factorial Series refers to the sum of the factorials of a given series of numbers. The factorial of a number is the product of all the numbers from 1 to that number. For example, the factorial of 5 is 5 x 4 x 3 x 2 x 1 = 120.
To find the Sum of Factorial Series, you need to first determine the factorial of each number in the series. Then, you simply add all the factorials together to get the sum. For example, if the series is 1, 2, 3, the sum of factorial series would be 1! + 2! + 3! = 1 + 2 + 6 = 9.
The formula for Sum of Factorial Series is n! + (n-1)! + (n-2)! + ... + 1!, where n is the number of terms in the series. This formula can be simplified to n! + (n-1)! + (n-2)! + ... + 1! = (n+1)! - 1.
The Sum of Factorial Series has many applications in mathematics, including in probability, combinatorics, and number theory. It is also used in the study of permutations and combinations, as well as in the calculation of binomial coefficients.
Some examples of Sum of Factorial Series are 1! + 2! + 3! = 1 + 2 + 6 = 9, 2! + 4! + 6! = 2 + 24 + 720 = 746, and 3! + 5! + 7! = 6 + 120 + 5040 = 5166. These examples show that the Sum of Factorial Series can be used to find the sum of factorials for any given series of numbers.