Find the first 3 nonzero terms of Maclaurin series, question on understanding the solution.

Question

zenmo · Answer

I circled the part that I don't understand with a blue circle, why is a 2 being added there?

zepdrix · Answer

Hmm <.<

zepdrix · Answer

Oh oh, because when you multiply these expansions together,
you also get x^4 from other combinations like x^2 * x^2.
There is no x^1 * x^3 in these series, so we don't have to worry about anything else.

Just the x^4 from the first series,
the x^4 from the second series,
and the x^2 from each of them multiplied together,
$\large\rm -x^2\cdot(-2x^2)$

zenmo · Answer

$$(-x^2)*(-2x^2)=2x^2$$ would that be correct?

zenmo · Answer

I'm probably missing something, I don't venture well in these areas. Haha

zepdrix · Answer

$$\large\rm -x^2\cdot(-2x^2)\quad=\quad 2x^{2+2}$$

zenmo · Answer

I gotten the same answer by doing the long way by doing $$(1-x^2+\frac{ 1 }{ 2 }x^4)(1-2x^2+\frac{ 2 }{ 3 }x^4)$$

zenmo · Answer

I still don't see how a 2 is being added for $$(\frac{ 2 }{ 3 }+2+\frac{ 1 }{ 2 })x^4$$

zenmo · Answer

I can see for ..x^2 by adding the two coefficients $$1-[(-1)+(-2)]x^3... = 1 -3x^3+..$$

zenmo · Answer

ops by x^3 i mean x^2

zepdrix · Answer

You don't have to expand this all the way out.$$\large\rm \left(1-x^2+\frac12x^4\right)\left(1-2x^2+\frac23x^4\right)$$We only want the x^4 powers that come out of this.
So we can cherry pick.

$\large\rm 1\cdot\frac23x^4$

$\large\rm \frac12x^4\cdot1$

$\large\rm -x^2\cdot(-2x^2)$

Those are the three multiplications which will give us x^4.

zepdrix · Answer

Are you a little rusty on exponent rule maybe? :o
Did you understand that multiplication I did earlier?

zenmo · Answer

I know how to do exponent rules

zepdrix · Answer

So those three multiplications give us,

$\large\rm \frac23x^4$

$\large\rm \frac12x^4$

$\large\rm 2x^4$

See where the 2 is coming from? :o

zenmo · Answer

Oh, I see it now, by doing the foil method or handing picking like you said, you will get 2x^4 from there

zenmo · Answer

So when you combine like terms, that is where the 2 comes from

zenmo · Answer

Thanks for the help! :)