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$$\sum_{n=1}^\infty \frac{5n^2+5n-3 \cdot2^n}{2^n\cdot n^2+2^n}$$
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$$\sum_{n=1}^\infty \frac{5n^2+5n-3\cdot 2^n}{2^n\cdot n^2+2^n\cdot n} =5\sum_{n=1}^\infty\frac{1}{2^n}-3\sum_{n=1}^\infty\frac{1}{n(n+1)}=5-3=2$$ |
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259. (ËҡʧÊÑÂÅͧ令鹴ÙàÃ×èͧ power series ´Ù¹Ð¤ÃѺ)
$$\begin{eqnarray} x+\frac{5x^3}{2\cdot3}+\frac{9x^5}{4\cdot5}+\frac{13x^7}{6\cdot7}+\ldots &=&x+(\frac{1}{2}+\frac{1}{3})x^3+(\frac{1}{4}+\frac{1}{5})x^5+(\frac{1}{6}+\frac{1}{7})x^7+\ldots\\ &=&(x+\frac{x^3}{3}+\frac{x^5}{5}+\ldots)+\frac{x}{2}(x^2+\frac{x^4}{2}+\frac{x^6}{3}+\ldots)\\ &=&\text{arctanh}\; x-\frac{x}{2}\ln{(1-x^2)}\\ \end{eqnarray}$$ 260. ¨Ò¡ $$(1-x)^{-2/3}=1+\frac{2}{3}x+\frac{2\cdot5}{3\cdot6}x^2 +\frac{2\cdot5\cdot8}{3\cdot6\cdot9}x^3+\ldots$$ àÁ×èÍãËé x=1/2 ¨Ðä´é¼ÅÃÇÁ͹ءÃÁ·Õèµéͧ¡Òä×Í $2^{2/3}$ |
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$$\int x\, d(\ln|1-x^2|) =\int \frac{2x^2}{x^2-1} \,dx$$ $$= \int2+ \frac{1}{x-1}- \frac{1}{x+1} \,dx$$ $$=2x+ \ln|x-1|- \ln|x+1| +C$$ $$=2x +\ln \left| \frac{x-1}{x+1} \right| +C$$ $$=2x+ \ln \left| \frac{1-x}{1+x} \right| +C$$
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$ \because \, c=-1 \ne1$
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¨Ò¡·Õè $$\left. \frac{ds}{dx} \right|_{x=0} =1$$ ´Ñ§¹Ñé¹àÁ×èÍ $x=0$ àÃҨеéͧä´éÇèÒ $$ -\frac12 \ln |1-x^2|+ 2(1-x^2)^{-1} +c=1$$ á¡éÊÁ¡ÒÃáÅéÇàÃҨоºÇèÒ $c=-1$ ¤ÃѺ
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