# Find radius and interval of convergence

Σ √n X (x-1)^(n)
n=1

## Homework Equations

Ratio Test

let the series Σ An be a series with positive terms and lim as n ---> infinity is an+1 / an = finite number

a.

the series converges if r < 1

b.

the series diverges if r > 1 or r is infinite

c.

the test is inconclusive if r = 1.

Root Test

http://ltcconline.net/greenl/courses/107/Series/ratio.htm

## The Attempt at a Solution

So far I've tried the Ratio test (can't figure out how to not get an indeterminate form of anyhthing less than infinity multiplied by zero)

Then the root test, I'm not even sure where to begin once I divide the n's by n.

-travis

## Answers and Replies

1) First use root test.
2) Then use ratio test

HallsofIvy
Science Advisor
Homework Helper
$$\frac{\sqrt{n+1}|x-1|^{n+1}}{\sqrt{n}|x-1|^n}= \sqrt{\frac{n+1}{n}}|x-1|= \sqrt{1+ \frac{1}{n}}|x-1|$$

What is the limit of that as n goes to infinity?

just to help out say the limit as n->inf is = X. (hint as HallsofIvy pointed out, with his answer, plug in inf into n and rationalize what happens to the function )

Then to find the radius of convergence by the ratio test, X<1
that implies that -1<X<1. The you need to get X with a coefficient 1, if not then divide both side.