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lfdahl
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Determine the sum:
\[\frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+...\]
\[\frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+...\]
The pattern in the series is that each term is increasing by a multiple of 1/4.
The number of terms to be evaluated is infinite, as the series continues indefinitely.
Yes, this series can be simplified by factoring out a common factor of 1/4, leaving us with 1/4(1+2+3+4+...).
The sum of the first 5 terms in the series is 5/2, as each term can be rewritten as 1/4(n+1) and when n is 1, 2, 3, 4, and 5, the sum is 5/2.
This series can be used in various applications such as calculating compound interest, growth rate of populations, and other exponential growth or decay scenarios.