You can't write a Maclaurin series (i.e., a Taylor series with a = 0) for f(x) = ln(x), since the function and all of its derivatives are not defined at x = 0. You can, however, write a Taylor series in powers of, say, x - 1, though.Austin said:I have just started learning about series and I don't see the benefit of shifting the series by using some "a" other than 0?
My textbook doesn't really tell the benefits it just says "it is very useful"'
Off the top of my head I don't know what the radius of convergence is for this series. You could write it in powers of x - e2, with e2 being about 7.39, which might be close enough to 9.Austin said:Would you be able to write a power series to estimate any value of lnx? I cannot think of a way to write a series to estimate ln9 for example... Is it possible? Every example I see is estimating ln2 and I feel like there must be some way to estimate other values but I cannot think of a way since the series diverges after 1 in each direction (from what I've seen)
Oh I see! The point of having a Taylor Series centered around some arbitrary a is to move your radius of convergence in a sense, is that correct?Mark44 said:Off the top of my head I don't know what the radius of convergence is for this series. You could write it in powers of x - e2, with e2 being about 7.39, which might be close enough to 9.
Something like that. You move the interval of convergence. The radius doesn't change.Austin said:Oh I see! The point of having a Taylor Series centered around some arbitrary a is to move your radius of convergence in a sense, is that correct?
The power series for ln((1+x)/(1-x)) converges for -1<x<1, which can (in principal) be used for any y = (1+x)/(1-x) > 0.Austin said:Would you be able to write a power series to estimate any value of lnx? I cannot think of a way to write a series to estimate ln9 for example... Is it possible? Every example I see is estimating ln2 and I feel like there must be some way to estimate other values but I cannot think of a way since the series diverges after 1 in each direction (from what I've seen)
A Taylor series generated at x=a is a mathematical representation of a function using a sum of terms, where each term is a derivative of the function evaluated at a specific point (x=a). It is a useful tool for approximating the behavior of a function near a particular point.
To calculate a Taylor series generated at x=a, one must first find the derivatives of the function at the point x=a. Then, these derivatives are substituted into the general form of a Taylor series, which is a sum of terms with coefficients determined by the derivatives. The more terms that are included in the series, the more accurate the approximation will be.
The point x=a is often chosen for a Taylor series because it makes the series simpler and easier to calculate. Additionally, the series will be most accurate near the point x=a, which is useful for approximating the behavior of the function near that point.
Yes, a Taylor series generated at x=a can be used to approximate a function at points other than x=a. However, the accuracy of the approximation will decrease as the distance from x=a increases. This is why choosing a point x=a that is close to the desired point of approximation is important.
One limitation of using Taylor series generated at x=a is that the series is only accurate near the point x=a. For points further away, the approximation may not be reliable. Additionally, the accuracy of the approximation depends on the smoothness of the function, so it may not work well for functions with discontinuities or sharp changes in behavior.