# Tagged Questions

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### Preventing “proof by homework”?

I am doing problem 3d in the Prologue of Spivak: Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$ I feel like my proof could pass off as ...
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### Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$

Let $n,m$ are positive integers satisfy the condition $n\ge m>0$ Prove that $\displaystyle\sum_{j=m}^n\sum_{k=0}^{2m}{4j\choose 2k}{2j-k\choose 2m-k}={2n+2m+1\choose 4m+1}2^{4m-1}$
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### Evaluate $\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$

Evaluate $$\lim_{n \to \infty} \sum_{j=0}^{n}{{j+n-1} \choose j}\frac{1}{2^{j+n}}$$ I don't understand where to start. Please help.
### Prove that $\lim_{n \to \infty} \binom{n}{k}a^n = 0$
I'm working with this problem but I have no idea how to solve it. Here $k$ is fixed and $0<a<1$. I was trying to use that $\lim_{n \to \infty} a^n =0$ and that $\binom{n}{k}\leq\frac{n^k}{k!}$ ...