A comparison on binomial expansion

• forumfann
In summary, the conversation discusses the inequality \sum_{k=n+1}^{2n}\left(\begin{array}{c}2n\\k\end{array}\right)x^{k}\left(1-x\right)^{2n-k}\leq2x for any x\in(0,1) and any positive integer n. One person suggests that the LHS is the "second half" of the terms in the expansion of (x+(1-x))^(2n)=1. However, another person notes that the inequality may not hold for small values of x. It is then mentioned that the LHS is always positive for x\in(0,1) and has been checked for various
forumfann
Could anyone help me on this question? Is it true that
$$\sum_{k=n+1}^{2n}\left(\begin{array}{c} 2n\\k\end{array}\right)x^{k}\left(1-x\right)^{2n-k}\leq2x$$
for any $$x\in(0,1)$$ and any positive integer $$n$$?

Any help on that will be greatly appreciated!

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Well, the LHS seems to be the "second half" of the terms of the expansion of ( x+ (1-x) )^(2n) = 1.

This gives that the LHS is $$1 - \sum_{k=0}^n \left(\begin{array}{c} 2n\\k\end{array}\right)x^{k}\left(1-x\right)^{2n-k}$$.

I can't see how that's incorrect, but then I note the inequality doesn't seem to hold anymore for when x is small, for the LHS would seem to tend to 1 whilst the RHS goes to zero. Maybe I'm going crazy..

Gib Z said:
the LHS would seem to tend to 1

When k=0, $$x^k(1-x)^{2n-k}$$ goes to 1 as x goes to 0, so $$1-\sum_{k=0}^{n}\left(\begin{array}{c} 2n\\ k\end{array}\right)x^{k}\left(1-x\right)^{2n-k}$$ tends to 0 as x goes to 0 because the other terms with k>0 has a factor x. But thanks anyway.

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Well if that is true, haven't you shown that the LHS is negative? I must have done something completely wrong as it seems my idea implies the original LHS must be negative..

Today obviously isn't my day. Sorry for wasting your time !

The LHS is always positive for $$x\in(0,1)$$. That is fine.

In fact, I have checked the cases of n=1 through 1000, and it holds for all the cases. But I still don't know how to show it in general. Any other help?

1. What is binomial expansion?

Binomial expansion is a mathematical method used to expand binomial expressions that have two terms raised to a certain power. It involves using the binomial theorem to find the coefficients of each term in the expansion.

2. How is binomial expansion useful?

Binomial expansion is useful in simplifying and solving complex algebraic equations, especially those involving polynomials. It is also used in probability and statistics to calculate the likelihood of certain events occurring.

3. What is the formula for binomial expansion?

The formula for binomial expansion is (a + b)^n = ∑ nCr * a^(n-r) * b^r, where n is the power, a and b are the two terms, and nCr is the combination formula for choosing r items from a set of n items.

4. Can binomial expansion be applied to more than two terms?

No, binomial expansion can only be applied to expressions with two terms. If there are more than two terms, it is considered a polynomial expansion and requires a different method of expansion.

5. What are some real-life applications of binomial expansion?

Binomial expansion is used in various fields such as finance, engineering, and physics. It is used to model and predict outcomes in stock markets, design and analyze bridges and buildings, and calculate the probabilities of outcomes in quantum mechanics.

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