Power series convergence

Question

anonymous · Answer

attachment

ganeshie8 · Answer

I is clearly wrong - recall the fact that harmonic series diverges but the alternating harmonic series converges

ganeshie8 · Answer

that eliminates first and third options

anonymous · Answer

Ok

ganeshie8 · Answer

II is trivially true because -2 lies in the interval |x|<3

anonymous · Answer

But, I thohught we only test abs(x)<1

ganeshie8 · Answer

the power series converges for x=3, and it is centered at 0, so the interval of convergence includes (-3, 3]

anonymous · Answer

III looks wrong since $$\Large -1<x <1$$
$$\Large -3<x<-1$$

ganeshie8 · Answer

III is also correct actually, shifting the center wil have no effect on radius of convergence

ganeshie8 · Answer

|x-2|<1

gives you

1<x<3

anonymous · Answer

Oh, ok

anonymous · Answer

Oh, how come I can't do what I did up there

-1<x<1
-3<x-2<-1    (I forgot the x-2)

Aren't I taking 2 from both sides?

ganeshie8 · Answer

thats an interesting question @doulikepiecauseidont

ganeshie8 · Answer

I'll try and answer this after sometime, need to think a bit... 
@dan815 @Kainui @TuringTest