Here are some pairs of equations. While one partner listens, the other partner should:

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

FromTheMoon [43]

Answer:

The Taylor series is $\ln(x) = \ln 3 + \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x-3)^n}{3^n n}$ .

The radius of convergence is $R=3$ .

Step-by-step explanation:

<em>The Taylor expansion.</em>

Recall that as we want the Taylor series centered at $a=3$ its expression is given in powers of $(x-3)$ . With this in mind we need to do some transformations with the goal to obtain the asked Taylor series from the Taylor expansion of $\ln(1+x)$ .

Then,

$\ln(x) = \ln(x-3+3) = \ln(3(\frac{x-3}{3} + 1 )) = \ln 3 + \ln(1 + \frac{x-3}{3}).$

Now, in order to make a more compact notation write $\frac{x-3}{3}=y$ . Thus, the above expression becomes

$\ln(x) = \ln 3 + \ln(1+y).$

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,

$\ln(x) = \ln 3 + \ln(1+y) = \ln 3 + \sum_{n=1}^{\infty} (-1)^{n+1} \frac{y^n}{n}.$

Now, substitute $\frac{x-3}{3}=y$ in the previous equality. Thus,

$\ln(x) = \ln 3 + \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x-3)^n}{3^n n}.$

<em>Radius of convergence.</em>

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

$R^{-1} = \lim_{n\rightarrow\infty} \sqrt[n]{|a_n|},$

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

In this case the coefficients of the Taylor series are

$a_n = \frac{(-1)^{n+1}}{ n3^n}$

and in consequence $|a_n| = \frac{1}{3^nn}$ . Then,

$\sqrt[n]{|a_n|} = \sqrt[n]{\frac{1}{3^nn}}$

Applying the properties of roots

$\sqrt[n]{|a_n|} = \frac{1}{3\sqrt[n]{n}}.$

Hence,

$R^{-1} = \lim_{n\rightarrow\infty} \frac{1}{3\sqrt[n]{n}} =\frac{1}{3}$

Recall that

$\lim_{n\rightarrow\infty} \sqrt[n]{n}=1.$

So, as $R^{-1}=\frac{1}{3}$ we get that $R=3$ .

8 0

4 years ago

7th Grade Math Help. I keep getting a different answer than from the ones on there.

netineya [11]

Answer:

100 apples are covered in sprinkles

7 0

3 years ago

Timothy drove his race car 10% farther using the gasoline from Smith's Station compared to gasoline from Jack's Station. After T

Fiesta28 [93]

Answer:

528

Step-by-step explanation:

x = 480

1.1 x = 480 * 1.1

480 + 48 = 528

5 0

4 years ago

I need some help pls:c

shtirl [24]

Answer:

The answer is x= 50 and y=50.

Step-by-step explanation:

So a straight line is always equal to 180 degrees. So, to find the length of the inner angle, subtract 100 from 180= 80. Now that you've gotten the value of that angle, you'll have to find x and y. All angles of a triangle add up to 180 degrees. 180-80= 100. The small dash on both of those lines means they are equal. Do 100/2 = 50. Hope this helps!

6 0

3 years ago

Sarah has 4 more $50 notes than $10 notes in her wallet.If the total amount of the notes is $380,how many $10 notes does Sarah h

Anni [7]

Answer:

3

Step-by-step explanation:

4 0

2 years ago