When solving an ODE by the power series method, one ends up with an expression multiplied by $x^n$ inside a summation equaling $0$, and the next step is to set that expression equal to $0$. For example, $$\sum_{n = 0}^{\infty} (na_{n + 1} + a_{n + 1} - 3a_n)x^n = 0$$ becomes $$na_{n + 1} + a_{n + 1} - 3a_n = 0.$$ Is there a geometric intuition behind what is going on with this step that makes it obvious why this is the right thing to do?
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3$\begingroup$ The series on the right side is $\sum 0x^n$. Since the series are equal, the coefficients are equal. $\endgroup$– B. GoddardCommented Jan 11, 2021 at 0:21
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1$\begingroup$ A polynomial is equivalent to $0$ if and only if all the coefficients are equal to $0$. Would you like a proof of this? $\endgroup$– Adam RubinsonCommented Jan 11, 2021 at 0:22
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