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Zarrin [17]
3 years ago
8

How many counterexamples do you need to prove a conjecture to be false

Mathematics
1 answer:
Hoochie [10]3 years ago
3 0

Answer:

one , i guess....

because the conjecture is only true if it satisfies all the cases.

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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
3 years ago
Sara can type 90 words in 4 minutes. About how many words would you expect her to type in 10 minutes at this rate?
Talja [164]

Answer:

She can type 225 words in 10 minutes

Step-by-step explanation:

We can find the rate by dividing the number of words by the time

90 words/ 4 minutes

22.5 words per minute

Now  to find the number of words in 10 minutes, multiply the rate times the time

22.5 * 10 = 225

She can type 225 words in 10 minutes

5 0
3 years ago
Read 2 more answers
A = 1/2(a+b)h <br> h=? <br> h= ? times ? / ?+?
slega [8]
H means multiplication
1/2 x (a+b) x h

h is height
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3 years ago
Help asap marking as BRAINLIST ace only
Soloha48 [4]

Answer:

O: The circle's name is "O"

Step-by-step explanation:

A circle is ALWAYS named for the center point. The point in the center of this circle is "O" so the circle is named "O"

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3 years ago
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(-5x^2)^3 =

= (-5)^3(x^2)^3

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