It looks like e x e^x

Calculus Level 3

If the value of ∑ n = 0 ∞ n 2 x n n ! \displaystyle\sum_{n=0}^{\infty}\frac{n^2x^n}{n!} is ( a x 2 + b x + c ) e x (ax^2+bx+c)e^x , where a a , b b , and c c are constants, enter the value of 123 a + 456 b + 789 c 123a+456b+789c .

Additional formula:

  • e x = ∑ n = 0 ∞ x n n ! \displaystyle e^x=\sum_{n=0}^\infty\frac{x^n}{n!}


The answer is 579.

Kenny Lau by Kenny Lau , 22, Hong Kong

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1 solution

Kenny Lau
Jul 24, 2015
  • ∑ n = 0 ∞ n 2 x n n ! \displaystyle \color{#D61F06}{\sum_{n=0}^{\infty}\frac{n^2x^n}{n!}}
  • = 0 2 x 0 0 ! + ∑ n = 1 ∞ n 2 x n n ! = \displaystyle \color{#EC7300}{\frac{0^2x^0}{0!}} + \color{#D61F06}{\sum_{n=1}^{\infty}\frac{n^2x^n}{n!}}
  • = 0 + ∑ n = 1 ∞ n 2 x n n ! = \displaystyle \color{#EC7300}0 + \color{#D61F06}{\sum_{n=1}^{\infty}\frac{n^2x^n}{n!}}
  • = 0 + ∑ n = 1 ∞ n x n ( n − 1 ) ! = \displaystyle \color{#EC7300}0 +\color{#D61F06}{\sum_{n=1}^{\infty}\frac{nx^n}{(n-1)!}}
  • = 0 + ∑ n = 0 ∞ ( n + 1 ) x n + 1 n ! = \displaystyle \color{#EC7300}0 + \color{#D61F06}{\sum_{n=0}^{\infty}\frac{(n+1)x^{n+1}}{n!}}
  • = 0 + x ∑ n = 0 ∞ ( n + 1 ) x n n ! = \displaystyle \color{#EC7300}0 + \color{#D61F06}{x\sum_{n=0}^{\infty}\frac{(n+1)x^n}{n!}}
  • = 0 + x ∑ n = 0 ∞ n x n n ! + x ∑ n = 0 ∞ x n n ! = \displaystyle \color{#EC7300}0 + \color{#3D99F6}{x\sum_{n=0}^{\infty}\frac{nx^n}{n!}} + \color{#69047E}{x\sum_{n=0}^{\infty}\frac{x^n}{n!}}
  • = 0 + x 0 x 0 0 ! + x ∑ n = 1 ∞ n x n n ! + x ∑ n = 0 ∞ x n n ! = \displaystyle \color{#EC7300}0 + \color{#20A900}{x\frac{0x^0}{0!}} + \color{#3D99F6}{x\sum_{n=1}^{\infty}\frac{nx^n}{n!}} + \color{#69047E}{x\sum_{n=0}^{\infty}\frac{x^n}{n!}}
  • = 0 + 0 + x ∑ n = 1 ∞ n x n n ! + x ∑ n = 0 ∞ x n n ! = \displaystyle \color{#EC7300}0 + \color{#20A900}0 + \color{#3D99F6}{x\sum_{n=1}^{\infty}\frac{nx^n}{n!}} + \color{#69047E}{x\sum_{n=0}^{\infty}\frac{x^n}{n!}}
  • = 0 + 0 + x ∑ n = 1 ∞ x n ( n − 1 ) ! + x ∑ n = 0 ∞ x n n ! = \displaystyle \color{#EC7300}0 + \color{#20A900}0 + \color{#3D99F6}{x\sum_{n=1}^{\infty}\frac{x^n}{(n-1)!}} + \color{#69047E}{x\sum_{n=0}^{\infty}\frac{x^n}{n!}}
  • = 0 + 0 + x ∑ n = 0 ∞ x n + 1 n ! + x ∑ n = 0 ∞ x n n ! = \displaystyle \color{#EC7300}0 + \color{#20A900}0 + \color{#3D99F6}{x\sum_{n=0}^{\infty}\frac{x^{n+1}}{n!}} + \color{#69047E}{x\sum_{n=0}^{\infty}\frac{x^n}{n!}}
  • = 0 + 0 + x 2 ∑ n = 0 ∞ x n n ! + x ∑ n = 0 ∞ x n n ! = \displaystyle \color{#EC7300}0 + \color{#20A900}0 + \color{#3D99F6}{x^2\sum_{n=0}^{\infty}\frac{x^n}{n!}} + \color{#69047E}{x\sum_{n=0}^{\infty}\frac{x^n}{n!}}
  • = 0 + 0 + x 2 e x + x e x = \displaystyle \color{#EC7300}0 + \color{#20A900}0 + \color{#3D99F6}{x^2e^x} + \color{#69047E}{xe^x}
  • = ( x 2 + x ) e x = \displaystyle (\color{#3D99F6}{x^2}+\color{#69047E}x)e^x
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