Proving order in series and sequences

**racewithferrari** · November 5th 2009, 04:26 AM

Show that $1^k + 2^k + 3^k + ... + (n-1)^k + n^k$ is ORDER $n^(k+1).$
(where k is a positive integer)

**chisigma** · November 5th 2009, 06:12 AM

The function of x...

$f(x) = x^{k} + (x-1)^{k} + (x-2)^{k} + \dots + (x-n+1)^{k}$ (1)

... is a polynomial of degree k and the coefficient of the term of degree k is $n$ , i.e. is...

$f(x)= n\cdot x^{k} + \dots \rightarrow f(x)=\mathcal {O} \{x^{k}\}$ (2)

If we set in (2) $x=n$ we have...

$f(n)= n^{k+1} + \dots \rightarrow f(n)=\mathcal {O} \{n^{k+1}\}$ (3)

Kind regards

$\chi$ $\sigma$

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