Question

Use the ratio test to find the radius of convergence of the power series

2x+16x^2+72x^3+256x^4+800x^5+...

R=

Answer 1

@agent0smith can you help please?

Answer 2

I think so. First find the r... divide one term by the term before it.

Answer 3

isnt it subtract not divide?

Answer 4

That'd be if it's arithmetic, it's def geometric - for one, the x goes to x^2 then x^3... and also arithmetics don't converge, so there is never a radius of convergence - ie these questions will NEVER be arithmetic sequences

Answer 5

when u divide the terms they arent the same so how do i know what r is?

Answer 6

Yeah i guess this isn't geometric.. i missed that it's a power series.

Answer 7

it happens! lol so then what do i do?

Answer 8

nth term: (2^n)(n^2)(x^n)

Answer 9

wats the nth term n how did u get it

Answer 10

I looked at the numbers 2, 16, 72, 256, 800. They did not have a common ratio and so it is not a geometric series. SO I started finding the prime factors of each number:
1: 2
16: 2 x 2 x 2 x 2
72: 2 x 2 x 2 x 3 x 3
256: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
800: 2 x 2 x 2 x 2 x 2 x 5 x 5

And looking for pattern I saw: (2^n) times (n^2)

Answer 11

so is that pattern our radius?

Answer 12

No that pattern gives you the general formula for the nth term of the series.

The nth term will be: 2^n times n^2 times x^n

Take the ratio of the (n+1)th term to the nth term and find the limit as n tends to infinity.

Answer 13

Working from the last row down to the first I recognized this pattern:

1st coefficient 2: 2 = 2^1 x 1^2
2nd coefficient 16: 2 x 2 x 2 x 2 = 2^2 x 2^2
3rd coefficient 72: 2 x 2 x 2 x 3 x 3 = 2^3 x 3^2
4th coefficient 256: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2^4 x 4^2
5th coefficient 800: 2 x 2 x 2 x 2 x 2 x 5 x 5 = 2^5 x 5^2
....
nth coefficient = 2^n x n^2

nth term of the series is: 2^n * n^2 * x^n

Answer 14

nth term of the series is: 2^n * n^2 * x^n

(n+1)th term of the series is: 2^(n+1) * (n+1)^2 * x^(n+1)

Take the ratio of (n+1)th term to the nth term:
\[\Large \frac{ (n+1)^{th} term }{ n^{th}term } = \]\[\Large \frac{ 2^{n+1}(n+1)^{2}x^{n+1}} { 2^{n}n^{2}x^{n} } = 2(\frac{ n+1 }{ n })^{2}x = 2(1 + \frac{ 1 }{ n })^{2}x\]
The limit of the ratio as n-->infinity is: 2x.
When we take the ratio it should be an absolute value so the limit is 2|x|

For the series to converge, the limit of the ratio as n-->infinity < 1

So 2|x| < 1. |x| < 1/2 or -1/2 < x < 1/2

The radius of convergence is 1/2

The interval of convergence is the interval from x = -1/2 to +1/2