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meriva
2 years ago
7

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 = 9
Mathematics
1 answer:
Bond [772]2 years ago
3 0

Taylor series is f(x) = ln2 + \sum_{n=1)^{\infty}(-1)^{n-1} \frac{(n-1)!}{n!(9)^{n}(x9)^{2}  }

To find the Taylor series for f(x) = ln(x) centering at 9, we need to observe the pattern for the first four derivatives of f(x). From there, we can create a general equation for f(n). Starting with f(x), we have

f(x) = ln(x)

f^{1}(x)= \frac{1}{x} \\f^{2}(x)= -\frac{1}{x^{2} }\\f^{3}(x)= -\frac{2}{x^{3} }\\f^{4}(x)= \frac{-6}{x^{4} }

.

.

.

Since we need to have it centered at 9, we must take the value of f(9), and so on.

f(9) = ln(9)

f^{1}(9)= \frac{1}{9} \\f^{2}(9)= -\frac{1}{9^{2} }\\f^{3}(x)= -\frac{1(2)}{9^{3} }\\f^{4}(x)= \frac{-1(2)(3)}{9^{4} }

.

.

.

Following the pattern, we can see that for f^{n}(x),

f^{n}(x)=(-1)^{n-1}\frac{1.2.3.4.5...........(n-1)}{9^{n} }  \\f^{n}(x)=(-1)^{n-1}\frac{(n-1)!}{9^{n}}

This applies for n ≥ 1, Expressing f(x) in summation, we have

\sum_{n=0}^{\infinite} \frac{f^{n}(9) }{n!} (x-9)^{2}

Combining ln2 with the rest of series, we have

f(x) = ln2 + \sum_{n=1)^{\infty}(-1)^{n-1} \frac{(n-1)!}{n!(9)^{n}(x9)^{2}  }

Taylor series is f(x) = ln2 + \sum_{n=1)^{\infty}(-1)^{n-1} \frac{(n-1)!}{n!(9)^{n}(x9)^{2}  }

Find out more information about taylor series here

brainly.com/question/13057266

#SPJ4

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Answer:

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Step-by-step explanation:

A bagel (Yiddish: בײגל, romanized: beygl; Polish: bajgiel; also historically spelled beigel)[1] is a bread product originating in the Jewish communities of Poland. It is traditionally shaped by hand into the form of a ring from yeasted wheat dough, roughly hand-sized, that is first boiled for a short time in water and then baked. The result is a dense, chewy, doughy interior with a browned and sometimes crisp exterior. Bagels are often topped with seeds baked on the outer crust, with the traditional ones being poppy and sesame seeds. Some may have salt sprinkled on their surface, and there are different dough types, such as whole-grain and rye.

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8 0
2 years ago
(7,5), (r,9) m=6<br><br> What is r?
Igoryamba

Answer:

The answer is that r = 7/6

Step-by-step explanation:

Given point: (7, 5)

Given slope: m = 6

Use the point slope form of the equation:

y - y1 = m(x - x1)

y - 5 = 6(x - 7)

y - 5 = 6x - 7

y = 6x + 2

Now, find r for the point (r, 9) by substitution:

9 = 6x + 2

6x + 2 = 9

6x = 7

x = 7/6

Proof:

y = 6x + 2

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= 42/6 + 2

= 7 + 2 = 9, giving (7/6, 9). r = 7/6

Hope this helps! Have a great day!

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