1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
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}+.....

You might be interested in
If the sum of a number and seven is tripled, the result is three less than twice the number. Find the number
hram777 [196]

Answer:

x = -24

Step-by-step explanation:

3(x + 7) = 2x - 3

3 * x = 3x

3 * 7 = 21

3x + 21 = 2x - 3

-21            -21

3x = 2x -24

-2x   -2x

x = -24

4 0
3 years ago
Felipe, Pedro y Miguel son fanáticos del cine. Felipe asiste cada 3 días; Pedro cada 6 días y Miguel cada 5 días. Los tres van j
lilavasa [31]

Answer:

Opción C, 30 días.

Step-by-step explanation:

Sabemos que:

Felipe asiste al cine cada 3 días

Miguel asiste cada 5 días.

Pedro asiste cada 6 días.

Los tres van juntos un día,  ¿dentro de cuántos días irán otra vez juntos?

Los 3 volverán a ir juntos luego de N días, tal que N es el menor múltiplo común de 3, 5 y 6.

Como 6 es múltiplo de 3, podemos ignorar al 3, y solo considerar al menor múltiplo en común de 5 y 6.

Además, como podemos reescribir estos números como un producto de números primos, así obtenemos:

6 = 2*3

5 = 5*1

Estos números no comparten factores entre ellos, por lo que el menor múltiplo entre 5 y 6 es simplemente 5*6 = 30

Entonces podemos concluir que N = 30

Felipe, Pedro y Miguel irán juntos al cine nuevamente dentro de 30 días.

La opción correcta es C.

5 0
3 years ago
How could the distance formula and slope be used to classify triangles and quadrilaterals in the coordinate plane?
S_A_V [24]

~ Use the distance formula to measure the lengths of the sides.

~ Use the slope to check whether sides are perpendicular and form right angles.

~ Use the slope to check whether the diagonals are perpendicular to each.

I hope this helps ^-^

7 0
3 years ago
Solve the system of linear equations using either substitution or the elimination method.
lara31 [8.8K]

Answer:

x = -1

y = 2

Step-by-step explanation:

x + y = 1

y = -x + 1

y = -x + 1  (Multiply this by -1)

y = 3x +5

-y = x - 1

y = 3x +5

0 = 4x + 4

4x = -4

x = -1

x + y = 1

-1 + y = 1

y = 2

4 0
3 years ago
No fakes please!<br> Ill give brainliest!!!
Andreas93 [3]

Step-by-step explanation:

1. 5/8 = 62.5%

2. 1/5 = 20%

3. 4 2/3 = 4.66%

8 0
3 years ago
Other questions:
  • Half of a number is three times the sum of a number ad five.
    9·1 answer
  • A group of employees bought a get-well gift for a co-worker. The gift cost $16.06. If there are 22 employees in the group, how m
    12·2 answers
  • Every 7 pages of an​ author's manuscript correspond to 2 published pages. How many published pages will a 182​-page manuscript​
    6·1 answer
  • What is the word form of the number 43.06?
    10·2 answers
  • PLEASE HELP! Since AB=1 and 1=BC, BC=AB by which property?
    10·1 answer
  • Which system has no solution?<br> Check all that appy.
    11·1 answer
  • If a number ends with 0, then it is divisible by 10. What is the contrapositive of this statement?
    8·1 answer
  • Four different ' four 4's problems ( where they all equal a different #) and four different four 4's that equal the same #
    9·1 answer
  • A square of side 72 cm and a rectangle of length 90 cm have same perimeter. Find the difference in areas ?
    14·2 answers
  • Simplify the monomial (Please show work)<br><br> (2ab^2)(4a^2b^3) - (10a^3b)(6b^4)
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!