Let S k - 1 k - 2 k - n k and Sm-n- r-1 m m-1 C r S r- Find S3-2

Given nbsp; S _ k = 1 ^ k + 2 ^ k +....+ n ^ k nbsp; nbsp; nbsp; nbsp;.....(1)Also given nbsp; S(m,n)=∑ _ r=1 ^ m _ nbsp; ^ m+1 C _ ...

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Integration Using Substitution

Let S k = 1...

Question

Let $S_{k} = 1^{k} + 2^{k} + . . . + n^{k}$ and $S (m, n) = \sum_{r = 1}^{m} {^{m + 1} C}_{r} S_{r}$ . Find $S (3, 2)$

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Solution

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S_{k} = 1^{k} + 2^{k} + 3^{k} + . . . . . + n^{k},

\sum_{r = 1}^{m}

^{(m + 1)} C_{r} S_{r}

s_{k} = 1^{k} + 2^{k} + 3^{k} . . . + n^{k},

\sum_{r = 1}^{m}^{m + 1} C_{r} S_{r} =, (n + 1)^{m + 1} - (n + 1) .

s_{4} .

S_{k} = sin (\frac{2 π k}{n + 1}) - i cos (\frac{2 π k}{n + 1})

P_{k} = cos (\frac{π}{2^{k}}) + i sin (\frac{π}{2^{k}}),

m = ∣ ∣ ∣ n \sum k = 1 S_{k} + \infty \prod k = 1 P_{k} ∣ ∣ ∣

2 m^{2}

S_{k} = sin (\frac{2 π k}{n + 1}) - i cos (\frac{2 π k}{n + 1})

P_{k} = cos (\frac{π}{2^{k}}) + i sin (\frac{π}{2^{k}}),

m = ∣ ∣ ∣ n \sum k = 1 S_{k} + \infty \prod k = 1 P_{k} ∣ ∣ ∣

2 m^{2}

{lim}_{n \to \infty} \frac{(1^{k} + 2^{k} + 3^{k} + . . . . . + n^{k})}{(1^{2} + 2^{2} + . . . . . + n^{2}) (1^{3} + 2^{3} + . . . . . + n^{3})} = F (k)

(k \in N)

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