# Find the values of #x# for which the following series is convergent?

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#\sum_(n=0)^(\infty)(2x-3)^n#

To determine the values of (x) for which a series converges, we typically need to examine the convergence of the series. Without knowing the specific series provided in the question, it's challenging to give a precise answer.

However, some common tests for convergence include the ratio test, the root test, the comparison test, and the integral test. Each of these tests can help determine the convergence behavior of a series for a given range of values of (x).

If you provide the specific series, I can offer more guidance on which convergence test to apply and how to determine the values of (x) for which the series converges.

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For Power Series, however, three cases are possible

So, here,

So, apply the Ratio Test:

Now, let's determine the interval:

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I made a mistake here, but the above answer has the same method and a correct answer, so just have a look at that instead.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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