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- Binomial Theorem for Negative Index When applying the binomial theorem to negative integers, we first set the upper limit of the sum to infinity; the sum will then only converge under specific conditions. Second, we use complex values of n to extend the definition of the binomial coefficient.
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The binomial theorem for positive integer exponents \( n \) can be generalized to negative integer exponents. This gives rise to several familiar Maclaurin series with numerous applications in calculus and other areas of mathematics.
In this explainer, we will learn how to use the binomial expansion to expand binomials with negative and fractional exponents.
2016年9月14日 · I have been trying to understand why the binomial theorem can work for negative and fractional indices. I understand that when raising binomials to positive integral indices, each coefficient is s...
2015年1月21日 · Comparing the formula for regular binomial expansion (n>1): $(a+b)^n=a^n + \binom{n}1a^{n-1}b + \binom{n}2a^{n-2}b^2 +...$ to binomial expansion for negative indices, (n<1): $(1+x)^n= 1 + nx + \
Understanding Binomial Expansion for Negative and Rational Index - Learn How to Expand and Simplify an Expression Step-by-Step!#mathsmadeeazywithpdy, #binomi...
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2011年11月26日 · The binomial expansion "really" sums from $0$ to $\infty$, not $0$ to $n$. In cases where $n$ is a nonnegative integer, then for $k>n$, we have that $\binom{n}{k}=0$. So terms past $k=n$ do not contribute, and you often see the upper limit "simplified" from $\infty$ down to $n$. $\endgroup$
Binomial Theorem for Negative Index. When applying the binomial theorem to negative integers, we first set the upper limit of the sum to infinity; the sum will then only converge under specific conditions. Second, we use complex values of n to extend the definition of the binomial coefficient.
