p=∑n=0∞x3n(3n)!, q=∑n=0∞x3n-2(3n-2)!, r=∑n=0∞x3n-1(3n-1)! then p + q + r =

p=∑n=0∞ x3n3n =1+x33+x66+....q=x1+x44+....r=x22+x55+....p+q+r=1+x1+x22+x33+.... =ex

p=∑n=0∞x3n(3n)!, q=∑n=0∞x3n-2(3n-2)!, r=∑n=0∞x3n-1(3n-1)! then p + q + r =

p=∑n=0∞ x3n3n =1+x33+x66+....q=x1+x44+....r=x22+x55+....p+q+r=1+x1+x22+x33+.... =ex

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$p = \sum_{n = 0}^{\infty} \frac{x^{3 n}}{(3 n)!}, q = \sum_{n = 0}^{\infty} \frac{x^{3 n - 2}}{(3 n - 2)!}, r = \sum_{n = 0}^{\infty} \frac{x^{3 n - 1}}{(3 n - 1)!}$ then $p + q + r =$

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$e^{3 x}$

$e^{x}$

$e^{2}$

$e^{2 x}$

answer is A.

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Detailed Solution

$\begin{array}{rcl} p & = & \sum_{n = 0}^{\infty} \frac{x^{3 n}}{3 n} \\ = & 1 + \frac{x^{3}}{3} + \frac{x^{6}}{6} + . . . . \\ q & = & \frac{x}{1} + \frac{x^{4}}{4} + . . . . \\ r & = & \frac{x^{2}}{2} + \frac{x^{5}}{5} + . . . . \\ p + q + r & = & 1 + \frac{x}{1} + \frac{x^{2}}{2} + \frac{x^{3}}{3} + . . . . \\ = & e^{x} \end{array}$