# Taylor series generated at x=a

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"'

Mark44
Mentor
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"'
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.

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 ive seen)

Mark44
Mentor
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 ive seen)
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.

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.
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
Mentor
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?
Something like that. You move the interval of convergence. The radius doesn't change.

mathman
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 ive seen)
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.

FactChecker
Another point: if you have a linear differential equation of the form $$y''+ f(x)y'+ g(x)y$$, with "initial values" y(a)= b, y'(a)= c; that is, with values of y and its derivative give at x= a, it is simplest to write the solution as a power series in powers of x-a. That is, in the form $$y= b+ c(x- a)+ p_2(x- a)^2+ p_3(x- a)^3+ \cdot\cdot\cdot$$ so that the first two coefficients are the given values, b and c. You can then write the functions f and g in Taylor series about a.