Find the taylor series for f(x) centered at the given value of a. [assume that f has a power series expansion. do not show that rn(x) → 0.] f(x) = ln(x), a = 3 f(x) = ln(3) + ∞ n = 1 find the associated radius of convergence r. r =

The Taylor series is .

The radius of convergence is .

Step-by-step explanation:

The Taylor expansion.

Recall that as we want the Taylor series centered at its expression is given in powers of . With this in mind we need to do some transformations with the goal to obtain the asked Taylor series from the Taylor expansion of .

Then,

Now, in order to make a more compact notation write . Thus, the above expression becomes

Notice that, if x is very close from 3, then y is very close from 0. Then, we can use the Taylor expansion of the logarithm. Hence,  

Now, substitute in the previous equality. Thus,

Radius of convergence.

We find the radius of convergence with the Cauchy-Hadamard formula:

Where stands for the coefficients of the Taylor series and for the radius of convergence.

In this case the coefficients of the Taylor series are

and in consequence . Then,

Applying the properties of roots

Hence,

Recall that

So, as we get that .

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step-by-step explanation: