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ANEK [815]
3 years ago
11

SOLUTION We observe that f '(x) = -1 / (1 + x2) and find the required series by integrating the power series for -1 / (1 + x2).

Mathematics
1 answer:
Ann [662]3 years ago
7 0

Answer:

Required series is:

\int{\frac{-1}{1+x^{2}} \, dx =-x+\frac{x^{3}}{3}-\frac{x^{5}}{5}+\frac{x^{7}}{7}+.....

Step-by-step explanation:

Given that

                           f'(x) = -\frac{1}{1 + x^{2}} ---(1)

We know that:

                  \frac{d}{dx}(tan^{-1}x)=\frac{1}{1+x^{2}} ---(2)

Comparing (1) and (2)

                           f'(x)=-(tan^{-1}x) ---- (3)

Using power series expansion for tan^{-1}x

f'(x)=-tan^{-1}x=-\int {\frac{1}{1+x^{2}} \, dx

= -\int{ \sum\limits^{ \infty}_{n=0} (-1)^{n}x^{2n}} \, dx

= -\sum{ \int\limits^{ \infty}_{n=0} (-1)^{n}x^{2n}} \, dx

=-[c+\sum\limits^{ \infty}_{n=0} (-1)^{n}\frac{x^{2n+1}}{2n+1}]

=C+\sum\limits^{ \infty}_{n=0} (-1)^{n+1}\frac{x^{2n+1}}{2n+1}

=C-x+\frac{x^{3}}{3}-\frac{x^{5}}{5}+\frac{x^{7}}{7}+.....

as

                 tan^{-1}(0)=0 \implies C=0

Hence,

\int{\frac{-1}{1+x^{2}} \, dx =-x+\frac{x^{3}}{3}-\frac{x^{5}}{5}+\frac{x^{7}}{7}+.....

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D=228

when 2 numbers or variables are next to eachother it means you need to multiply. So in this case r=57 and t=4 and we mulitply. 57 x 4 = 228.  So the answer is 228. Hope this helps and have a great day!

8 0
3 years ago
someone please help :( a sample of the amount of rent paid for one-bedroom apartments of similar size near the University of Ore
Volgvan

Answer:

Oregon:

Median: 467.5

Mean: 451.33

Standard Deviation: 113.6

Washington:

median: 380

Mean: 396.6

Standard deviation: 56.3

Step-by-step explanation:

To calculate the median, we need to organize the price and select the value that is in the middle position.

To calculate the mean, we need to sum all the values and divide them by the number of values

To calculate the standard deviation, we need to square the distances between every price and the mean, sum it all and divide them by the number of values less 1 and finally calculate the square root.

So, for Oregon, the organized values are:

$295, $345, $460, $475, $538, $595

The median is a value between $460 and $475. it is calculated as:

\frac{460+475}{2}=467.5

The mean is:

\frac{295+475+345+595+538+460}{6}=451.33

The standard deviation is:

\sqrt{\frac{(295-451.3)^2+(475-451.3)^2+(345-451.3)^2+(595-451.3)^2+(538-451.3)^2+(460-451.3)^2}{6-1}}=113.6

At the same way, we can calculate the median, mean, and standard deviation for Washington and get:

median: 380

Mean: 396.6

Standard deviation: 56.3

3 0
3 years ago
There are 1,760 yards in a mile. How many full laps would Danny have to run around the block to run a mile?
oee [108]

Answer:

how many yards is the track

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2 years ago
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7 0
3 years ago
ASAP:::::: 75 POINTS TO THE BRAINLIEST!!!!!
Misha Larkins [42]

Answer:

t = 6 4/25 hours

Step-by-step explanation:

The formula to find out how long it takes them to do the job is

1/m  + 1/ a  = 1 /t

where m is the time for Matthew to do the job

a is the time for Arianna to do the job

and t is the time for them to do the job together

m = 14 hours

a = 11 hours

Substituting what we know

1/14 + 1/11 = 1/t

Multiplying by 154t (14*11*t) so we can clear the fractions

154t*(1/14 + 1/11) = 1/t* 154t

 11t + 14t = 154

Combine like terms

25t = 154

Divide by 25

t = 154/25

t = 6 4/25 hours


6 0
3 years ago
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