Binomial expansion, how to do them quickly?
I'm using this formula for my binomial expansion, but this takes a long time, especially to calculate the binomial coefficient (we aren't allowed to use a calculator). Consider the following (easy) example: …
Sum of 'the first $k$' binomial coefficients for fixed $N$
May 30, 2025 · One can take this a step further. In addition to combining pairs of terms of the original sum N choose i to get a sum of terms of the form N+1 choose 2j+c, where c is always 0 or always 1, …
Expansion of 1/sqrt(1-x) - Math Help Forum
Apr 15, 2012 · How do you find the binomial expansion of 1/sqrt(1-x) in series form? I know what the term by term expansion is but i'm trying to find the series representation, The closest i have found …
Binomial expansion of $ (1-x)^n$ - Mathematics Stack Exchange
In my opinion, this substitution is the best way to see "how" to get the binomial expansion, as the OP originally asked, because it demonstrates a method which reduces the problem to the expression OP …
Binomial Expansion for non-commutative setting - MathOverflow
What could be a reference about binomial expansions for non-commutative elements? Specifically, where can I find a closed formula for the expansion of $ (A+B)^n$ where $ [A,B]=C$ and $ [C,A]= …
Negative Exponents in Binomial Theorem - Mathematics Stack Exchange
Nov 26, 2011 · Negative Exponents in Binomial Theorem Ask Question Asked 14 years, 5 months ago Modified 5 years, 3 months ago
Upper limit on the central binomial coefficient - MathOverflow
The simplest upper bound to prove is $4^n$ (which is still stronger than your bound) and just follows from the binomial expansion of $ (1+1)^ {2n}$. Peter's answer gives a less wasteful estimate.
Binomial expansion for $ (x+a)^n$ for non-integer n
Binomial expansion for $ (x+a)^n$ for non-integer n Ask Question Asked 10 years, 4 months ago Modified 1 year, 7 months ago
binomial coefficients - Sum of the first m terms of the expansion $ (x ...
Jan 19, 2022 · Sum of the first m terms of the expansion $ (x+y)^n$ Ask Question Asked 4 years, 2 months ago Modified 4 years, 2 months ago
combinatorics - Understanding the binomial expansion for negative …
Sep 14, 2016 · 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 …