Using comparison tests and limit comparison test

Sunwoo Bae
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Homework Statement
Determine whether the following series converge or diverge by using appropriate test. (Picture below)
Relevant Equations
None
CB722E56-DDC4-4368-8164-96578AEE2EF1.jpeg

The answer sheet states that the series converges by limit comparison test (the second way).
In the case of this particular problem, would it be also okay to use the comparison test, as shown above? (The first way)

Thank you!
 
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Using the comparison test is fine here.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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