# Prove at least one of this identities using binomial theorem. [closed]

I've tried to prove at least one of them, but nothing is working. Could you give me a hint? Also i'd like to know something about properties of $\sum$, like multiplying them, etc(links).

## closed as off-topic by John Doe, Matthew Conroy, Namaste, Rolf Hoyer, KrishNov 26 '17 at 4:46

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• Hint: For (7) and (8), consider the binomial expansion of $(x+1)^n$ and take derivatives. Then evaluate at $x = 1$. – eepperly16 Nov 25 '17 at 21:51

All of these identities can be derived from $$(1+x)^n = \sum_{k=0}^{n}\binom{n}{k}x^k. \tag{1}$$ For example, check how $(1)$ can be used to prove $5)$ here.
For $6)$, integrate $(1)$ as $$\int_{0}^1(1+x)^n dx = \frac{2^{n+1}-1}{n+1}=\sum_{k=0}^{n}\binom{n}{k}\int_{0}^{1}x^kdx=\sum_{k=0}^{n}\frac{\binom{n}{k}}{k+1}. \tag {2}$$
Likewise, take derivatives of $(1)$ w.r.t. $x$ to get $7)$ and $8)$.