**Find The Interval Of Convergence Of The Series** – Sometimes we are asked about the radius and region of convergence of a Taylor series. To find these things, we first need to find a power series representation for the Taylor series.

If we represented the Taylor series as a power series, we identify ???a_n??? In the_??? and substitute it into the limit formula in the ratio test to determine where the series converges.

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## Find The Interval Of Convergence Of The Series

Using the graph below, find the cubic Taylor series around ???a=3??? for ???f(x)=ln(2x)???. Then find the power series representation of the Taylor series as well as the radius and interval of convergence.

## Solved Find The Radius Of Convergence R Of The Series.

Since we have already completed the graph, the value in the right column gives the coefficient of each term in the Taylor polynomial in the form

We want to find a power series representation for the Taylor series above. The first thing we can see is that the exponent of each ???(x-3)??? is equal to ???n??? Value of this term, what this means

Will be part of the power series representation. The fractional coefficient before ???(x-3)??? Conditions can be represented by

Finally we have to deal with the negative sign???n=2??? cope with Expression. If we multiply our terms by

### Solved Find The Interval Of Convergence For The Series.

N=2??? term will be negative and ???n=1??? and ???n=3??? The conditions will be positive. Note that none of these generalizations for ???n=0??? applies. Therefore, we will remove this term from the power series representation.

To find the radius of convergence, we identify ???a_n??? In the_??? with the power series representation we just found.

Because we have the indefinite form ???infty/infty??? When we try to evaluate the limit, we divide the numerator and the denominator by the variable with the highest degree to reduce the fraction.

To find the region of convergence, we take the inequality we used to find the radius of convergence and solve for x.

#### Question Video: Finding The Radius Of Convergence For The Maclaurin Series Of A Trigonometric Function

Because the ratio test tells us that the series converges if ???L<1???, so we find the inequality.

Because the inequality has the form ???|x-a|<R??? has, we can say that the radius of convergence ???R=3??? is.

To find the region of convergence, we take the inequality we used to find the radius of convergence and solve for ???x??? on

We need to test the endpoints of the inequality by connecting them to the power series representation. We start with ???x=0???.

#### Solved Find The Radius Of Convergence And Interval Of

We have shown that the series at ???x=0??? diverged. and converges to ???x=6???, which means that the interval of convergence is

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