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zh3f
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[tex]sum_{i=k}^{n} {i \choose k}i^{-t}[/tex]
where t is a constant.
Does it have a closed form?
where t is a constant.
Does it have a closed form?
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A binomial coefficient is a mathematical term that represents the number of ways to choose a subset of items from a larger set. It is denoted by the symbol "n choose k" and can be calculated using the formula n! / (k! * (n-k)!), where n and k are positive integers.
Sums involving binomial coefficients are commonly used in mathematics to represent the number of ways to arrange or choose objects from a larger set. They can also be used to calculate probabilities in various scenarios, such as in coin tosses or card games.
One example of a sum involving binomial coefficients is the binomial theorem, which states that (a+b)^n = sum from k=0 to n of (n choose k) * a^(n-k) * b^k. This formula is used in algebra to expand binomial expressions.
Pascal's triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. The coefficients in a sum involving binomial coefficients can be found in each row of Pascal's triangle, and the sum itself can be calculated by adding the numbers in the row.
Yes, sums involving binomial coefficients have many real-world applications. They are commonly used in probability and statistics to calculate the likelihood of certain events occurring. They can also be used in fields such as genetics and economics to model and analyze different scenarios.