N乘以X的N次方求和 ,N从1到无穷大
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N乘以X的N次方求和 ,N从1到无穷大
![N乘以X的N次方求和 ,N从1到无穷大](/uploads/image/z/19178939-11-9.jpg?t=N%E4%B9%98%E4%BB%A5X%E7%9A%84N%E6%AC%A1%E6%96%B9%E6%B1%82%E5%92%8C+%2CN%E4%BB%8E1%E5%88%B0%E6%97%A0%E7%A9%B7%E5%A4%A7)
an=n*x^n
Sn=1*x^1+2*x^2+3*x^3+.+(n-1)x^(n-1)+n*x^n
xSn=x^2+2x^3+3x^4+.+(n-1)x^n+n*x^(n+1)
Sn-xSn=x+x^2+x^3+x^4+.+x^n-n*x^(n+1)
=x(1-x^n)/(1-x) -n*x^(n+1)
=(x-x^(n+1)/(1-x)-n*x^(n+1)
=(x-x^(n+1)-nx^(n+1)+nx^(n+2))/(1-x)
=(x-(1+n)x^(n+1) +nx^(n+2))/(1-x)
Sn=(x-(1+n)x^(n+1)+nx^(n+2)/(1-x)^2
(1) 若x>=1 则和为无穷大
(2)若-1
Sn=1*x^1+2*x^2+3*x^3+.+(n-1)x^(n-1)+n*x^n
xSn=x^2+2x^3+3x^4+.+(n-1)x^n+n*x^(n+1)
Sn-xSn=x+x^2+x^3+x^4+.+x^n-n*x^(n+1)
=x(1-x^n)/(1-x) -n*x^(n+1)
=(x-x^(n+1)/(1-x)-n*x^(n+1)
=(x-x^(n+1)-nx^(n+1)+nx^(n+2))/(1-x)
=(x-(1+n)x^(n+1) +nx^(n+2))/(1-x)
Sn=(x-(1+n)x^(n+1)+nx^(n+2)/(1-x)^2
(1) 若x>=1 则和为无穷大
(2)若-1
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