Why and how in the following expression $$ y_{n+1}=y_n+hy^{\prime}_n+\frac{1}{2}\left[ \frac{y^{\prime}_{n+1}-y^{\prime}_n}{h}+O(h) \right]h^2+O(h^3) $$ $$\Rightarrow y_{n+1}=y_n+h\left( y^{\prime}_n+\frac{1}{2}y^{\prime}_{n+1}+\frac{1}{2}y^{\prime}_n \right)+O(h^3) $$ the $O(h)$ term eliminated?
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2Because you are multiplying $h^2$ terms of order $O(h)$ which is of $O(h^3).$ – Ehsan M. Kermani Sep 12 '14 at 15:04
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We distribute the product to find $O(h) \cdot h^2 = O(h^3)$, which gets absorbed into the other $H(h^3)$ term – Ben Grossmann Sep 12 '14 at 15:13
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OMG! Why am I stupid! – Dante Sep 12 '14 at 15:27
