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lfdahl
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Prove the following identity:\[\sum_{n =1}^{\infty }\frac{1}{\binom{n+r}{r+1}}=\frac{r+1}{r},\: \: \: \: r,n \in \mathbb{N}.\]
The Binomial coefficient challenge is a mathematical problem that involves finding the number of combinations of a given set of objects. It is commonly used in probability and statistics to calculate the likelihood of certain events occurring.
The binomial coefficient, also known as the "choose" function, is calculated using the formula nCr = n! / (r!(n-r)!), where n is the total number of objects and r is the number of objects being chosen.
The binomial coefficient has many applications in mathematics, including in probability, statistics, and combinatorics. It is also used in fields such as genetics, physics, and computer science.
No, the binomial coefficient is only defined for integer values of n and r. However, there are extensions of the formula for non-integer values, such as the Gamma function.
Pascal's triangle is a visual representation of the binomial coefficients, where each number in the triangle corresponds to the coefficient for a specific n and r value. The triangle can be used to quickly find the binomial coefficient for a given n and r without having to use the formula.