This week, a teacher of mine wrote this down. He's been wrong before and so I've been looking at it and trying to see how this is true, but I can't seem to see how left leads to right. Is this correct?
$$\sum_{i=0}^{n} b_{2i}x^{2i} + \sum_{i=1}^{n} b_{2i-1}x^{2i-1} = \sum_{i=0}^{2n} b_{i}x^{i} $$
A quick way of checking whether summations are correct is by putting the values of $i$ and working out the terms of either sides of the equation. Here, First sum can be expanded as: $b_{0} + b_{2} x^2 + b_{4} x^4 + ... $
Second sum can be expanded as: $b_{1} x + b_{3} x^3 + b_{5} x^5 + ... $
So indeed, adding them will give us: $b_{0} + b_{2} x^2 + b_{4} x^4 ... + b_{1} x + b_{3} x^3 + b_{5} x^5 ...$
On Rearranging, we get: $b_{0} + b_{1}x^1 + b_{2}x^2 ... = \sum_{i=0}^{2n} b_{i}x^{i}$ which is the desired sum.