The answer is: for Largest: [?3, 7). and for Smallest: [0, 4). I HAVE NO IDEA HO

Question

The answer is: for Largest: [?3, 7). and for Smallest: [0, 4).

I HAVE NO IDEA HOW THESE WERE FOUND AND NO IDEA THE PROCCESS TO FIND THEM. I know how to use the ratio test to see if it converges to what value of x, but no idea how to look at this problem when they give you values of x for when it converges / diverges. Can you please, please, please, please, please, please, please (im saying so many times because i'm so irritated by having posted this question 3 times to here already and everytime the answer is wrong/ and so terrible answered) and also I've emailed countless people and gotten no help. SO PLEASE IF YOU KNOW HOW TO DO THIS DON'T THINK I CAN UNDERSTAND YOUR EXPLANATION IF YOU SKIP ANYTHING BECAUSE I WONT SO DONT SKIP OVER ANYTHING. ALSO IF YOU CAN TELL I WILL DOWNVOTE DOWNVOTE DOWNVOTE IF YOU JUST SKIP OVER MY WARNING AS IM BEYOND IRRITATED THAT THIS SIMPLE OF A PROBLEM IS GIVING SO MANY PEOPLE A HARD TIME. I WILL UPVOTE UPVOTE UPVOTE IF YOU CAN ACTUALLY PRODUCE SOMETHING WORTH UNDERSTANDING.

Thank you, (if done correctly.)

3. If bn(z-2)" converges at x = 0 but diverges at x = 7, what is the largest possible interval of convergence of this series? What's the smallest possible?

Explanation / Answer

sum of bn(x 2)^n

the centre of the series is 2

let R be radius of convergence

==>|x-2|<R

given the series converges for x=0

if it converges at x = 0, the radius must be at least 2 which is the distance between the center and x = 0.

|x-2|<2

-2<(x-2)<2

0<x<4

smallest interval is[0,4) , we donot know weather series converges at x=4

diverges at x = 7

distance from centre to 7 is 7-2=5 units==>radius canbe atmost 5

|x-2|<5

-5<x-2<5

-3<x<7

===>largest possible interval of convergence is (-3,7), given series diverges for x=7 ,series may conerge or diverge for x=-3

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