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thereddevils
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How is
[tex]1+p+\frac{p^2}{2!}+\frac{p^3}{3!}+...=e^p[/tex] ?
[tex]1+p+\frac{p^2}{2!}+\frac{p^3}{3!}+...=e^p[/tex] ?
thereddevils said:How is
[tex]1+p+\frac{p^2}{2!}+\frac{p^3}{3!}+...=e^p[/tex] ?
Mentallic said:I know it has to do with taylor expansions, but I've never studied this so I can't answer your question. I'd also like to see a proof for this so this is like some pointless post I'm making so I can subscribe to this thread
The binomial expansion is a mathematical formula that allows us to expand (a + b)^n, where n is a non-negative integer, into a series of terms. It is used to simplify polynomial expressions and to solve problems involving combinations and probabilities.
The binomial expansion is related to e^p through the general term of the expansion, which is given by (n choose k) * a^(n-k) * b^k. This general term can be rewritten as (n choose k) * (a/b)^k * b^n-k. As n approaches infinity, (a/b)^k approaches e^p, which is why the binomial expansion is often used to approximate e^p.
The binomial expansion is used in probability to calculate the probability of getting a certain number of successes in a series of trials. The expansion allows us to calculate the number of possible outcomes for a given number of successes, which is then divided by the total number of possible outcomes to obtain the probability.
Yes, the binomial expansion can be used for non-integer powers using the generalized binomial theorem. This theorem extends the expansion to include real or complex exponents, and it is often used in calculus and other advanced mathematical concepts.
The binomial expansion has many applications in fields such as statistics, finance, and physics. For example, it can be used to calculate the probability of a certain number of successes in a series of coin flips, to approximate compound interest in financial investments, and to model the trajectory of a projectile in physics.