In this question , k , n are non-negative integers and k ≦ n

$\dpi{150} \\i) Prove : k\binom{n}{k}=n\binom{n-1}{k-1}\\\\Hence , show that : \frac{k-1}{n(n-1)}\binom{n}{k}=\frac{1}{k}\binom{n-2}{k-2}\\\\ ii) Evaluate : \sum_{k=0}^{n}\binom{n}{k} and \sum_{k=0}^{n}(-1)^k\binom{n}{k}\\\\ Hence or otherwise, find a closed form for the sum \\\\ \sum_{k=0}^{n}\frac{(-1)^k}{k+1}\binom{n}{k} and \sum_{k=0}^{n}\frac{(-1)^k}{k+2}\binom{n}{k}$

one more hint for the last 2 series

you can use the changing variable(s) of a summation

$\sum_{k=r}^{n}f(k)=\sum_{k=r+m}^{n+m}f(k-m)$

This type of question would be more common in AL Pure Maths than in DSE.

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Is it a dse level [email protected]@

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use integration for the first sum

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