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

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

##### 2 Answers

#### Explanation:

When trying to determine the radius and/or interval of convergence of power series such as these, it is best to use the Ratio Test, which tells us for a series

If

If

If

For Power Series, however, three cases are possible

a. The power series converges for all real numbers; its interval of convergence is

b. The power series converges for some number

c. The most frequent case, the power series converges for

So, here,

So, apply the Ratio Test:

So, if

Now, let's determine the interval:

We need to plug

Therefore, the series converges for

We can use the ratio test which says that if we have a series

it is definitely convergent if:

In our case,

So, we need to check when

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.