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I am having trouble with finding a formula of the multiplication 3 formula power series.

\[ \left(\sum_{n=0}^{\infty} a_nx^n \right)\left(\sum_{k=0}^{\infty} b_kx^k \right)\left(\sum_{m=0}^{\infty} c_mx^m \right) \]

Work:

For the constant term:

$a_0b_0c_0$

For The linear term : $(a_1 b_0 c_0 + a_0 b_1 c_0 + a_0 b_0 c_1)x$ + $a_0b_0c_0$

For the quadratic term: $a_2 b_2 c_2 x^6 + a_2 b_2 c_1 x^5 + a_2 b_1 c_2 x^5 + a_1 b_2 c_2 x^5 + a_2 b_2 c_0 x^4 + a_2 b_1 c_1 x^4 + a_1 b_2 c_1 x^4 + a_2 b_0 c_2 x^4 + a_1 b_1 c_2 x^4 + a_0 b_2 c_2 x^4 + a_2 b_1 c_0 x^3 + a_1 b_2 c_0 x^3 + a_2 b_0 c_1 x^3 + a_1 b_1 c_1 x^3 + a_0 b_2 c_1 x^3 + a_1 b_0 c_2 x^3 + a_0 b_1 c_2 x^3 + a_2 b_0 c_0 x^2 + a_1 b_1 c_0 x^2 + a_0 b_2 c_0 x^2 + a_1 b_0 c_1 x^2 + a_0 b_1 c_1 x^2 + a_0 b_0 c_2 x^2 + a_1 b_0 c_0 x + a_0 b_1 c_0 x + a_0 b_0 c_1 x + a_0 b_0 c_0$

I am seeing that the indexes are summing up to the power of x. But how to say that in the indexes?

Thanks,

Cbarker1