Series help. *queston attached below* will give medal

Question

itiaax · Answer

So I need some help with number 2 in this picture. I've found the common ratio to be 1/2x, is this correct? If it is correct, I am having problems finding the value for which the series is convergent.

haichi · Answer

Dont know sorry

ganeshie8 · Answer

x/2 is the common ratio here, what do you know about convergence of geometric series ?

itiaax · Answer

I know that -1<r<1 if the series is convergent

ganeshie8 · Answer

plug the common ratio there and solve x

ganeshie8 · Answer

-1 < x/2 < 1

itiaax · Answer

-2<x<2

ganeshie8 · Answer

that means the series converges when x is between -2 and 2

itiaax · Answer

yes

ganeshie8 · Answer

what do they mean by $\large S_2$ in part 2 ?

itiaax · Answer

The sum of the second term, I'm guessing

ganeshie8 · Answer

im not sure @SithsAndGiggles

anonymous · Answer

$S_2$ probably denotes the second partial sum, i.e. $\displaystyle\sum_{n=1}^2\frac{x^n}{2^{n-1}}$

itiaax · Answer

Yes ^

ganeshie8 · Answer

yeah there seems to be a typo i think.. that inequality should be $S_2 \lt  4$

ganeshie8 · Answer

because sum of first terms would be less than 4 when the series converges
$$x+\frac{x^2}{2} \lt  4$$ when -2<x<2

itiaax · Answer

Hmm, let me process this