Prove that .nC1−(1+21).nC2+(1+21+31).nC3+…+(−1)n−1(1+21+31+….+n1).nCn=n1

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Prove that .^{{{n}}}{C}_{{{1}}}-{({1}+{{1}}{{2}})}.^{{{n}}}{C}

Question

Prove that $.^{n} C_{1} - (1 + \frac{1}{2}) .^{n} C_{2} + (1 + \frac{1}{2} + \frac{1}{3}) .^{n} C_{3} + \dots + (- 1)^{n - 1} (1 + \frac{1}{2} + \frac{1}{3} + \dots . + \frac{1}{n}) .^{n} C_{n} = \frac{1}{n}$

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Prove that

.^{n} C_{1} - (1 + \frac{1}{2}) .^{n} C_{2} + (1 + \frac{1}{2} + \frac{1}{3}) .^{n} C_{3} + \dots + (- 1)^{n - 1} (1 + \frac{1}{2} + \frac{1}{3} + \dots . + \frac{1}{n}) .^{n} C_{n} = \frac{1}{n}

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Question Text

Prove that

.^{n} C_{1} - (1 + \frac{1}{2}) .^{n} C_{2} + (1 + \frac{1}{2} + \frac{1}{3}) .^{n} C_{3} + \dots + (- 1)^{n - 1} (1 + \frac{1}{2} + \frac{1}{3} + \dots . + \frac{1}{n}) .^{n} C_{n} = \frac{1}{n}