Discover the Power of Maclaurin's Series: Solving 1/(1+x^2) for -1 > x > 1

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ThereforeS = 1/(1-y)In summary, the conversation is about finding a Maclaurin series for 1/(1+x^2) and it is derived using the binomial series. It can also be obtained by substituting x^2 with y in the geometric series 1/(1-y).
  • #1
newton1
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1/(1+x^2)=1-x^2 + x^4-x^6+...+(-1)^n(x^2n)+... -1 > x > 1
how to get this??...
 
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  • #2
It's basically Maclaurin's series.

f(x) = f(0) + x*f'(0) + (x^2)/2! * f''(0) +...

Which derives the binomial series:
(1+a)^n = 1+ na + n(n-1)/2! * a^2 + n(n-1)(n-2)/3! * a^3 ... As long as |a|<1

Substitute a = x^2 and n = -1 et viola!
 
  • #3


Originally posted by Newton1
1/(1+x^2)=1-x^2 + x^4-x^6+...+(-1)^n(x^2n)+... -1 > x > 1
how to get this??...

If you do not want to go all the way to the general theory of series of powers (Taylor and MacLaurin series), you can simply use the result of the geometric series

1/(1-y)=1+y+y^2+...+y^n+...

which can be obtained with basic arithmetic arguments and substistute y with -x^2.
 
  • #4
Yes. Because, if you abbreviate the right hand side as S, then obviously
y*S = S-1
 

1. What is Maclaurin's series?

Maclaurin's series is a special case of the Taylor series, which is a mathematical representation of a function as an infinite sum of its derivatives at a single point. Maclaurin's series specifically refers to a Taylor series centered at x = 0.

2. How do you find the Maclaurin series of a function?

To find the Maclaurin series of a function, you need to calculate the derivatives of the function at x = 0. Then, substitute these values into the general formula for a Taylor series, which is f(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + ...

3. What is the significance of Maclaurin's series?

Maclaurin's series is useful in mathematics and engineering because it allows us to approximate complicated functions with a simpler polynomial. This can make calculations and analysis easier and more efficient.

4. What is the difference between Maclaurin's series and Taylor series?

The main difference between Maclaurin's series and Taylor series is the center point at which the series is centered. Maclaurin's series is centered at x = 0, while Taylor series can be centered at any point in the function's domain.

5. Can Maclaurin's series be used for all functions?

No, Maclaurin's series can only be used for functions that are infinitely differentiable at x = 0. This means that the function must have derivatives of all orders at that point. Otherwise, the series will not converge and cannot be used to approximate the function.

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