Bernoulli number

[Soopreecha]

กำหนดให้
$$\frac{x}{e^x-1}=
1-\frac{x}{2}+\frac{B_1x^2}{2!}-\frac{B_2x^4}{4!}+\frac{B_3x^6}{6!}-...$$


พิสูจน์
$$ 1^p+2^p+3^p+...+n^p=
\frac{n^{p+1}}{p+1}+\frac{n^p}{2}+\frac{B_1pn^{p-1}}{2!}-\frac{B_2p(p-1)(p-2)n^{p-3}}{4!}+...$$
ทำยังไงครับ

[Anton]

Problem. Let $B_1,B_2,B_3,\ldots$ be the real numbers such that $$\frac{x}{\mathrm{e}^x-1}=1-\frac{x}{2}+\frac{B_1}{2!}\,x^2-\frac{B_2}{4!}\,x^4+\frac{B_3}{6!}\,x^6-\ldots\,.$$ Show that $$1^p+2^p+3^p+\ldots+n^p=\frac{n^{p+1}}{p+1}+\frac{n^p}{2}+\frac{B_1}{2!}\,p\,n^{p-1}-\frac{B_2}{4!}\,p(p-1)(p-2)\,n^{p-3}+\ldots\,.$$

Let $S^p_n$ denote the sum $1^p+2^p+3^p+\ldots+n^p$ for each positive integer $n$ and for each nonnegative integer $p$. Observe that
$$f_n(x):=\sum_{p=0}^\infty\,\frac{S^p_n}{p!}\,x^p=\sum_{p=0}^\infty\,\sum_{k=0}^n\,\frac{k^p}{p!}\,x^p=\sum_{k=1}^n\,\sum_{p=0} ^\infty\,\frac{(kx)^p}{p!}\,.$$
Therefore,
$$f_n(x)=\sum_{k=1}^n\,\exp(kx)=\frac{\big(\exp(nx)-1\big)\,\exp(x)}{\exp(x)-1}\,.$$
Consequently,
$$f_n(x)=\left(\frac{\exp(nx)-1}{x}\right)\,\left(\frac{x}{\exp(x)-1}\right)\,\exp(x)\,.$$
Now, we have
$$\exp(x)=\sum_{k=0}^\infty\,\frac{1}{k!}\,x^k\,,$$
$$\frac{\exp(nx)-1}{x}=\sum_{r=0}^\infty\,\frac{n^{r+1}}{(r+1)!}\,x^r\,,$$
and
$$\frac{x}{\exp(x)-1}=-\frac{x}{2}-\sum_{s=0}^\infty\,\frac{(-1)^s}{(2s)!}\,B_{s}\,x^{2s}\,,$$
where $B_0,B_1,B_2,\ldots$ are the Bernoulli numbers (with $B_0:=-1$). Thus, the coefficient of $x^p$ in $f_n(x)$ is given by the sum
$$s_n^p:=-\frac{1}{2}\,\sum_{r=0}^{p-1}\,\frac{n^{r+1}}{(r+1)!\,(p-1-r)!}+\sum_{\substack{r,s\geq 0\\ r+2s\leq p}}\,\frac{(-1)^s\,B_{s}\,n^{r+1}}{(r+1)!\,(2s)!\,(p-r-2s)!}\,.$$
That is,
$$\begin{align}S_n^p&=p!\,s_n^p\\&=\sum_{\ell=0}^{p+1}\,\frac{p!}{(p-\ell+1)!}\,n^{p+1-\ell}\,\left(-\sum_{s=0}^{\left\lfloor \frac{\ell}{2}\right\rfloor}\,\frac{(-1)^s\,B_s}{(2s)!\,(\ell-2s)!}-\frac{1}{2\,(\ell-1)!}\right)\,.\tag{$\star$}\end{align}$$
We need to verify that
$$-\sum_{s=0}^{\left\lfloor \frac{\ell}{2}\right\rfloor}\,(-1)^s\,\binom{\ell}{2s}\,B_s-\frac{\ell}{2}=\begin{cases}0&\text{if }\ell\geq 3\text{ is odd}\,,\\
-(-1)^{\frac{\ell}{2}}\,B_{\frac{\ell}{2}}&\text{if }\ell\geq 2\text{ is even}\,.
\end{cases}\tag{*}$$
For convenience, write
$$b_s:=\begin{cases}
-(-1)^{\frac{s}{2}}B_{\frac{s}{2}}&\text{if }s\geq 0\text{ is even}\,,\\
-\dfrac{1}{2}&\text{if }s=1\,,\\
0&\text{if }s\geq 3\text{ is odd}\,.
\end{cases}$$
That is, $\dfrac{x}{\exp(x)-1}=\sum\limits_{s=0}^\infty\,\dfrac{b_s}{s!}\,x^s$. Hence, (*) is equivalent to
$$\sum_{s=0}^\ell\,\binom{\ell}{s}\,b_s=b_\ell\text{ for }\ell=2,3,4,\ldots\,.\tag{#}$$

We shall now prove (#). From $\dfrac{x}{\exp(x)-1}=\sum\limits_{s=0}^\infty\,\dfrac{b_s}{s!}\,x^s$, we get
$$x=\big(\exp(x)-1\big)\sum\limits_{s=0}^{\infty}\,\dfrac{b_s}{s!}\,x^s=\left(\sum_{r=1}^{\infty}\,\frac{x^r}{r!}\right)\,\left(\sum\limits_{s=0} ^{\infty}\,\dfrac{b_s}{s!}\,x^s\right)\,.$$
This yields
$$x=\sum_{\ell=0}^\infty\,\frac{x^\ell}{\ell!}\,\sum_{s=0}^{\ell-1}\,\binom{\ell}{s}\,b_s\,.$$
This shows that
$$\sum_{s=0}^{\ell-1}\,\binom{\ell}{s}\,b_s=0\text{ for }\ell=2,3,4,\ldots\,.$$
This is equivalent to (#).

Therefore, using ($\star$), we see that $S_n^0=n$ and for an integer $p\geq 1$, we have
$$S_n^p=\frac{n^{p+1}}{p+1}+\frac{n^p}{2}-\sum_{\ell=1}^{\left\lfloor\frac{p}{2}\right\rfloor}\,\frac{(-1)^\ell\,p(p-1)(p-2)\cdots (p-2\ell+2)}{(2\ell)!}\,B_\ell\,n^{p+1-2\ell}\,.$$
Thus, the claim is established. Here are some examples:
$$S_n^1=\frac{n^2}{2}+\frac{n}{2}=\frac{n(n+1)}{2}\,,$$
$$S_n^2=\frac{n^3}{3}+\frac{n^2}{2}+\frac{n}{6}=\frac{n(n+1)(2n+1)}{6}\,,$$
$$S_n^3=\frac{n^4}{4}+\frac{n^2}{2}+\frac{n}{4}=\left(\frac{n(n+1)}{2}\right)^2\,,$$
$$S_n^4=\frac{n^5}{5}+\frac{n^4}{2}+\frac{n^3}{3}-\frac{n}{30}=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}\,,$$
and
$$S_n^5=\frac{n^6}{6}+\frac{n^5}{2}+\frac{5n^4}{12}-\frac{n^2}{12}=\frac{n^2(n+1)^2(2n^2+2n-1)}{12}$$

[share]

Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in number theory.

The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent
and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers,
in the Euler-Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

Wiki

[kongp]

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