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#1
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͹ءÃÁ͹ѹµì ªèÇ·դÃѺ
\sum_{n = 1}^{\infty}[sin (n\pi - \pi /2) + (-1)¡¡ÓÅѧ n+1 ·Ñé§ËÁ´Êèǹ (-1) ¡¡ÓÅѧ n+2 ¤Ù³ -3]·Ñé§ËÁ´Â¡¡ÓÅѧ n
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#2
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§§¡Ñºâ¨·Âì¤ÃѺẺ¹ÕéäËÁ¤ÃѺ
$$\sum_{n=1}^{\infty } [\dfrac{\sin (n\pi-\frac{\pi}{2})+(-1)^{n+1}}{(-3)(-1)^{n+2}}]^n$$
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#3
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#4
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#5
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ÍéÍ äÁèà»ç¹äà ¢Íº¤Ø³ ¤ÃÒºººº
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#6
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ÍéÒ§ÍÔ§:
$\displaystyle\therefore\sum_{n=1}^{\infty } [\dfrac{\sin (n\pi-\frac{\pi}{2})+(-1)^{n+1}}{(-3)(-1)^{n+2}}]^n=\sum_{n=1}^{\infty } \left(\frac{2}{3}\right)^n=2$ |
#7
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äÁèÂÒ¡ÍÂèÒ§·Õè¤Ô´¤ÃѺ á»Å§ à·ÍÁ·ÕèÁÕ $sine$ ãËéÍÍ¡¡çä´éáÅéÇÅèФÃѺ
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#8
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¼Á¹Ö¡ÍÍ¡áÅéǤÃѺ
Å᷹ͧ¤èÒ $$n=1;[\dfrac{\sin (\pi-\frac{\pi}{2})+1}{(-3)(-1)}]^1=\frac{2}{3}$$ $$n=2;[\dfrac{\sin (2\pi-\frac{\pi}{2})-1}{(-3)(1)}]^2=(\frac{2}{3})^2$$ $$n=3;[\dfrac{\sin (3\pi-\frac{\pi}{2})+1}{(-3)(-1)}]^3=(\frac{2}{3})^3$$ $$.$$ $$.$$ $$.$$ ä´éÇèÒ $$\sum_{n=1}^{\infty} [\dfrac{\sin (n\pi-\frac{\pi}{2})+(-1)^{n+1}}{(-3)(-1)^{n+2}}]^n=\frac{2}{3}+(\frac{2}{3})^2+(\frac{2}{3})^3+...$$ $$S_{\infty }=\frac{2}{3}+(\frac{2}{3})^2+(\frac{2}{3})^3+...$$ ¨Ò¡Êٵà $S_{\infty}=\dfrac{a_1}{1-r}$ ä´éÇèÒ $a_1=\frac{2}{3} $ áÅÐ $r=\frac{2}{3}$ ´Ñ§¹Ñé¹ $$S_{\infty }=\frac{2}{3}+(\frac{2}{3})^2+(\frac{2}{3})^3+...=\dfrac{\frac{2}{3}}{1-\frac{2}{3}}=2$$ $$\therefore \sum_{n=1}^{\infty} [\dfrac{\sin (n\pi-\frac{\pi}{2})+(-1)^{n+1}}{(-3)(-1)^{n+2}}]^n=2$$ »Å.äÁè·ÃÒºÇèÒ¶Ù¡äËÁ¹Ð¤ÃѺ
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#9
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