Proving the Root Test
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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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The root test is proven assuming ratio test.
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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 ...
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