<s>[INST] Evaluate the sum ${1 - \frac{1}{2} {n \choose 1} + \frac{1}{3} {n \choose 2} + \ldots + (-1)^n \frac{1}{n+1} {n \choose n}}$ Evaluate the sum $${1 - \frac{1}{2} {n \choose 1} + \frac{1}{3} {n \choose 2} + \ldots + (-1)^n \frac{1}{n+1} {n \choose n}}.$$
I have tried comparing this to the similar problem here.
I believe I need to differentiate or integrate? But I'm not sure how that might work.
Any ideas? Thanks.
 [/INST] This is $$\sum_{k=0}^{n}\frac{(-1)^{k}}{k+1}\binom{n}{k}.$$ Then multiplying each term by $\frac{n+1}{n+1},$ we get $$\sum_{k=0}^{n}\frac{(-1)^{k}}{n+1}\binom{n+1}{k+1}=\frac{1}{n+1}\sum_{k=1}^{n+1}(-1)^{k-1}\binom{n+1}{k}.$$ Adding and subtracting $1$ and applying the Binomial Theorem gives $$\frac{1}{n+1}-\frac{1}{n+1}\sum_{k=0}^{n+1}\binom{n+1}{k}(-1)^{k}1^{n+1-k}=\frac{1}{n+1}(1-(1-1)^{n+1})=\frac{1}{n+1}.$$
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