Urgend Geometric series question

[assyrian_77]

First of all, $\sum_{j=0}^{\infty}(-1)^{j}x^{2j} \neq x^{2} + x^{4} + x^{6} + x^{8}$, it is $\sum_{j=0}^{\infty}(-1)^{j}x^{2j}=1-x^{2} + x^{4} - x^{6} + x^{8}...$.

 

[arildno]

No. You MUST learn to read properly and stop writing sloppy and nonsensical maths.

We have:
$$(-1)^{n}x^{2n}\sum_{j=0}^{\infty}(-1)^{j}x^{2j}=\frac{(-1)^{n}x^{2n}}{1+x^{2}}$$

 

[assyrian_77]

And second; you know also this: $\frac{1}{1-q}=1 + q + q^2 + q^3 +...$.

Can you see the connection now?

I strongly suggest you to over the entire problem again and - as arildno is saying - read it properly.

 

[Mathman23]

No. You MUST learn to read properly and stop writing sloppy and nonsensical maths.

We have:
$$(-1)^{n}x^{2n}\sum_{j=0}^{\infty}=\frac{(-1)^{n}x^{2n}}{1+x^{2}}$$

Then
$$\sum_{j=0}^{\infty}(-1)^{j}x^{2j}=1-x^{2} + x^{4} - x^{6} + x^{8} +\cdots + (-1)^{n}x^{2n} = \sum_{j=0}^{\infty}\frac{(-1)^{n}x^{2n}}{1+x^{2}}$$

 

[assyrian_77]

We have:
$$(-1)^{n}x^{2n}\sum_{j=0}^{\infty}=\frac{(-1)^{n}x^{2n}}{1+x^{2}}$$

Sorry arildno, but that doesn't make sense.

 

[arildno]

Sorry arildno, but that doesn't make sense.

You're right, it doesn't I'll fix it right away.