Theoretical/non-tedious question about Frobenius method

[s3a]

When using the Frobenius method of solving differential equations using power series solutions, I get a solution

y = (indicial_stuff) + (infinite_summation_stuff) = 0

for a differential equation differential_stuff = 0.

WHY is it that I can say

(indicial_stuff) = 0?

If

y = (indicial_stuff) * (infinite_summation_stuff) = 0

(NOTICE THE MULTIPLICATION INSTEAD OF ADDITION) then it would make sense to me that

(indicial_stuff) = 0

but with the addition, I cannot make sense of this.

I tried to generalize this question and hope that I haven't made it more confusing but, if I have made the question confusing, please tell me and I will clear things up as best as I can.

Basically, I'm just asking for the reason why this – the isolation of the indicial polynomial equation (and the other summation portion) - is justified.

Any input would be greatly appreciated!
Thanks in advance!

 

[s3a]

The answer to my question was: "by equating coefficients."