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What if absolute convergence test gives the result of 'inconclusive' for a given power series?

We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know which of three condition (1. converges only for center a, 2. converges for all x, 3. converges within the radius R ) the power series comes under. Well, definitely 1st is valid since its a subset of the other two, but we cannot confirm 2nd ,3rd condition I think.

We just know it converges/diverges. So.. definitely in this case 3nd condition is not true because it is valid only when there exists R.

Then..is it automatically condition 1 or 2 always valid?

What if absolute convergence test gives the result of 'inconclusive' for a given power series?

We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know which of three condition (1. converges only for center a, 2. converges for all x, 3. converges within the radius R ) the power series comes under. Well, definitely 1st is valid since its a subset of the other two, but we cannot confirm 2nd ,3rd condition I think.

We just know it converges/diverges. So.. definitely in this case 3nd condition is not true because it is valid only when there exists R.

Then..is it automatically condition 1 or 2 always valid?

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