binomial exponential and logarithmic series pdf

Binomial Exponential And Logarithmic Series Pdf

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3 2 World Problem Practice Logarithmic Functions

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In mathematics , the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor who introduced them in If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series , after Colin Maclaurin , who made extensive use of this special case of Taylor series in the 18th century. Taylor polynomials are approximations of a function, which become generally better as n increases.

More precisely:. This is essentially equivalent to Euler's definition of the Gamma function :. Part ii follows from formula 5 , by comparison with the p-series. To prove iii , first use formula 3 to obtain. This completes the proof of iii. The usual argument to compute the sum of the binomial series goes as follows.

Binomial Sums

When they exist, the recurrence equations that give solutions to these equations can be generated quickly using Zeilberger's algorithm. For , the closed-form solution is given by. Franel , was the first to obtain recurrences for ,. Riordan , p. Franel , was also the first to obtain the recurrence for ,.

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The coefficients, called the binomial coefficients, are defined by the formula. The theorem is useful in algebra as well as for determining permutations and combinations and probabilities. For positive integer exponents, n , the theorem was known to Islamic and Chinese mathematicians of the late medieval period. Isaac Newton discovered about and later stated, in , without proof, the general form of the theorem for any real number n , and a proof by John Colson was published in

Logarithmic Function: The function that is the inverse of an exponential function is called a logarithmic function which is denoted by log b x.

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