Linked Questions

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Partial Sum of a Polynomial [duplicate]

Possible Duplicate: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$ I'm searching for a way to find the partial sum of a polynomial, is there any way of doing this ...
4answers
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3answers
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Find the value of a succession of additions

$$1^2+2^2+3^2+...+10000$$ How do you find the exact value of that? I'm studying induction, and I'm still not sure how to get that value.
6answers
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I am just trying to calculate $$\lim_{n\to\infty} \frac{1^4+2^4+\dots+n^4}{1^4+2^4+\dots+n^4+(n+1)^4}.$$ To do this I apply formula for sum of fourth powers of $n$ number. My result: $$\lim_{n\to\... 2answers 203 views Identity with Bernoulli numbers: \sum\limits_{k=1}^{n}k^p=\frac{1}{p+1}\sum\limits_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j} How I can prove that$$\sum_{k=1}^{n}k^p=\frac{1}{p+1}\sum_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j},$$where B_j is the jth Bernoulli number? I hope to find the answer. Thanks for help. 6answers 94 views Find the sum to n terms$$S=1^2+3^2+6^2+10^2+15^2+.......$$My attempt is as follows:$$T_n=\left(\frac{n\cdot\left(n+1\right))}{2}\right)^2T_n=\frac{n^4+n^2+2\cdot n^3}{4}S=\frac{1}{4}\cdot\sum_{n=1}^{n}\left(...

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