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Find the coefficient of x^n in (1+x/(1!)...

Find the coefficient of `x^n` in `(1+x/(1!)+(x^2)/(2!)++(x^n)/(n !))^2` .

Text Solution

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We have
`(1+(x)/(1!) + (x^(2))/(2!) + "....." + (x^(n))/(n!))^(2)`

We observe that `x^(n)` term is generated if term of the brackets are multiplied such that sum of exponent is n.
Hence, the coefficient of `x^(n)` is
`1 xx (1)/(n!) + (1)/((n-1)!) + (1)/(2!) xx (1)/((n-2)!) + "...." + (1)/(n)`
`= 1/(n!) ((n!)/(n!) + (n!)/((b-1)!1!) + (n!)/((n-2)!2!) + "...." + (n!)/(n!))`
`= (1)/(n!) (.^(n)C_(0) + .^(n)C_(1) + .^(n)C_(2) + "....." + .^(n)C_(n))`
`= (2^(n))/(n!)`
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