How do you find the taylor series for #ln(1+x^2)#?
Convergent when
which is the series we were looking for.
By signing up, you agree to our Terms of Service and Privacy Policy
To find the Taylor series for ( \ln(1+x^2) ), you start by finding the derivatives of ( \ln(1+x^2) ) with respect to ( x ) and evaluating them at ( x = 0 ) to find the coefficients of the series. The formula for the Taylor series expansion of a function ( f(x) ) about ( x = a ) is:
[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n ]
For ( \ln(1+x^2) ), the first few derivatives are:
[ f(x) = \ln(1+x^2) ] [ f'(x) = \frac{2x}{1+x^2} ] [ f''(x) = \frac{2(1-x^2)}{(1+x^2)^2} ] [ f'''(x) = \frac{-8x}{(1+x^2)^3} ]
Evaluating these derivatives at ( x = 0 ), we get:
[ f(0) = \ln(1) = 0 ] [ f'(0) = \frac{0}{1} = 0 ] [ f''(0) = \frac{2}{1} = 2 ] [ f'''(0) = \frac{0}{1} = 0 ]
So, the Taylor series expansion for ( \ln(1+x^2) ) about ( x = 0 ) is:
[ \ln(1+x^2) = 0 + 0(x-0) + \frac{2}{2!}(x-0)^2 + 0(x-0)^3 + \cdots ]
[ = x^2 + \frac{x^4}{2} + \frac{x^6}{3} + \cdots ]
Therefore, the Taylor series for ( \ln(1+x^2) ) is:
[ \ln(1+x^2) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^{2n}}{n} ]
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- How do you use differentiation to find a power series representation for #1/(6+x)^2#?
- What is the interval of convergence of #sum_1^oo [(3x)^n(x-2)^n]/(nx) #?
- How do you find the Maclaurin series of the function #f(x)=5cos(10x^2)#?
- How do you find the radius of convergence #Sigma (x^n)/(5^(nsqrtn))# from #n=[0,oo)#?
- How do you find the Taylor polynomial Tn(x) for the function f at the number a #f(x)=sqrt(3+x^2)#, a=1, n=2?
- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7