# Proving the Root Test

A) Prove root test " lim(sqrt|An)|)=L as n goes to infinity" assuming ratio test "lim(|An+1)|/|A n|)=L as n goes to infinity"

ps. {An} is a sequence of non-zero complex numbers

b) Prove that although the following power series have R=1 sum(nz^n) does not converge on any point of the unit circle.

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Problem:

a) Prove root test " lim(sqrt|An)|)=L as n goes to infinity" assuming ratio test "lim(|An+1)|/|A n|)=L as n goes to infinity"

ps. {An} is a sequence of non-zero complex numbers

b) Prove that although the following power series have R=1

sum(nz^n) does not converge on any point of the unit circle

Solution:

a) First off all, I have to make a correction to the above statement:

it is not sqrt|An| under the limit, but |An|^(1/n) or in other words, it is not "square" root, but root of order "n" (or fractional power).

Now, we can rephrase the theorem which we ...

#### Solution Summary

The root test is proven assuming ratio test.