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I know for $n=2,$ it's $1$, for $n=3$, it's $1/4$. But how to generalize and then prove? Pls let me know if you're unable to see the image attached. enter image description here

Let $k\geq2$ be a natural number. Find the value of $\sum_{n=1}^{\infty} \dfrac{1}{n(n+1)(n+2)..(n+k-1)}$

Show using above result that $\sum_{n=1}^{\infty} \dfrac{1}{n^k}$ converges for k=2,3,4,..

Saby123
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Bhart Lal
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  • At least for smaller $k$ the first part (c) should be solvable using telescopic sums, see https://math.stackexchange.com/questions/293244/compute-sum-k-1n-frac-1-kk-1 for $k=2$ and https://math.stackexchange.com/questions/1109391/calculate-s-sum-k-1n-frac-1kk1k2 for $k=3$. – DominikS Oct 02 '23 at 14:55
  • See https://math.stackexchange.com/q/1004004/42969 – Martin R Oct 02 '23 at 14:58
  • @MartinR thanks for that. How could I solve the b part? – Bhart Lal Oct 02 '23 at 16:15

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