What is the perimeter of the triangle below? Round to the nearest hundredth.

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do n

aliya0001 [1]

Answer:

$f(x)=\sum_{n=1}^{\infty}(-1)^{(n-1)}2^{n}\dfrac{x^n}{n}$

Step-by-step explanation:

The Maclaurin series of a function f(x) is the Taylor series of the function of the series around zero which is given by

$f(x)=f(0)+f^{\prime}(0)x+f^{\prime \prime}(0)\dfrac{x^2}{2!}+ ...+f^{(n)}(0)\dfrac{x^n}{n!}+...$

We first compute the n-th derivative of $f(x)=\ln(1+2x)$ , note that

$f^{\prime}(x)= 2 \cdot (1+2x)^{-1}\\f^{\prime \prime}(x)= 2^2\cdot (-1) \cdot (1+2x)^{-2}\\f^{\prime \prime}(x)= 2^3\cdot (-1)^2\cdot 2 \cdot (1+2x)^{-3}\\...\\\\f^{n}(x)= 2^n\cdot (-1)^{(n-1)}\cdot (n-1)! \cdot (1+2x)^{-n}\\$

Now, if we compute the n-th derivative at 0 we get

$f(0)=\ln(1+2\cdot 0)=\ln(1)=0\\\\f^{\prime}(0)=2 \cdot 1 =2\\\\f^{(2)}(0)=2^{2}\cdot(-1)\\\\f^{(3)}(0)=2^{3}\cdot (-1)^2\cdot 2\\\\...\\\\f^{(n)}(0)=2^n\cdot(-1)^{(n-1)}\cdot (n-1)!$

and so the Maclaurin series for f(x)=ln(1+2x) is given by

$f(x)=0+2x-2^2\dfrac{x^2}{2!}+2^3\cdot 2! \dfrac{x^3}{3!}+...+(-1)^{(n-1)}(n-1)!\cdot 2^n\dfrac{x^n}{n!}+...\\\\= 0 + 2x -2^2 \dfrac{x^2}{2!}+2^3\dfrac{x^3}{3!}+...+(-1)^{(n-1)}2^{n}\dfrac{x^n}{n}+...\\\\=\sum_{n=1}^{\infty}(-1)^{(n-1)}2^n\dfrac{x^n}{n}$

3 0

2 years ago

PLEASE HELP ME!A MILLION THANKS TO THE PEOPLE THAT ANSWERED ME!!.NO LINKS.WILL GIVE BRAIN IF YOU ANSWER CORRECTLY!THANK YOU SOO

azamat

Answer:

What eventually led to conflicts between settlers and American Indians in Virginia?

farmers destroying the soil with tobacco crops

farmers moving onto land close to the coast

indentured servants encroaching on settlers’ land

colonial traders charging too much for tobacco

Step-by-step explanation:

vWhat eventually led to conflicts between settlers and American Indians in Virginia?

farmers destroying the soil with tobacco crops

farmers moving onto land close to the coast

indentured servants encroaching on settlers’ land

colonial traders charging too much for tobacco

5 0

2 years ago

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It takes Carl 2/3 of an hour to clean his room every day.How many hours did he spend cleaning his room in the month of May?

MrMuchimi

20 2/3 hrs or 20 hours and 40 minutes cleaning his room. There are 30 months in may, so you multiply this by 2/3 to get 20 2/3 hrs

7 0

3 years ago

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-3 on a number line

babymother [125]

-3 on a number line would look like 3 right from 0.

Let's just say we have a number line. A number line is a line that represents numbers. I'll say the positive side is on the top/left and the negative side (-3) on the bottom/right. -3 is a negative number. There is the sign (-) before 3. That indicates it's a negative number. Since it's a negative and 3, we move 3 down or right.

6 0

2 years ago

What percent of 68 is 41?

EastWind [94]

It will be 165.85365853658536. Hope this helps !!

8 0

2 years ago

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