Using Standard Taylor Series to build other Taylor Series

In summary, the provided notes explain how the Taylor series for ln(1 + x) and ln(1 - x) can be used to approximate ln t for any positive real number t. The first paragraph discusses the formula for ln \left ( \dfrac{1 + x}{1 - x} \right ) and its domain, while the second paragraph compares the domains of ln(1 + x) and ln(1 - x) and the accuracy of their approximations. It is also shown that every positive real number t can be expressed in the form \dfrac{1 + x}{1 - x} for some x in the range -1 < x < 1.
  • #1
lyd123
13
0
Hello there, I am studying Taylor series, and in the slides given to us we calculated the taylor series of ln $(\frac{1+x}{1-x} )$ = ln(1 + x) − ln(1 − x), by using standard Taylor series of ln(1 + x).

The notes then proceed to say :
" It can be shown that every positive real number t can be expressed in the form
t =$\frac{1+x}{1-x} $ , for some x in the range −1 < x < 1.
Thus the Taylor series in this activity can be used to find an approximation for ln t for any t in the domain (0, ∞) of ln.

In contrast, the series for ln(1 + x) can be used to find approximations for ln t only for
$0 < t \le2$ since these are the only values of t that can be expressed in the form
t = 1 + x for some x in the range $0 < t \le1$. For both series, the further x is from 0, the more terms of the series
have to be evaluated in order to obtain the desired accuracy "

The second paragraph makes sense to me, but in the first paragraph, how can be it be true that every positive real number t can be expressed in the form
t =$\frac{1+x}{1-x} $ for some x in the range −1 < x < 1.
for eg. if t=20

Thank you in advance for any help. I think maybe I am missing something obvious.
 
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  • #2
lyd123 said:
Hello there, I am studying Taylor series, and in the slides given to us we calculated the taylor series of ln $(\frac{1+x}{1-x} )$ = ln(1 + x) − ln(1 − x), by using standard Taylor series of ln(1 + x).
I've never heard of the arguments stated in the rest of this. Interesting.

One point, though. You can represent \(\displaystyle ln \left ( \dfrac{1 + x}{1 - x} \right ) = ln(1 + x) - ln(1 - x)]\), but only if the Taylor series for ln(1 + x) and ln(1 - x) converge. Otherwise their sum may not be unique, and might not even be equal to the series for \(\displaystyle ln \left ( \dfrac{1 + x}{1 - x} \right )\).

-Dan
 
  • #3
Hi lyd123,

Given a positive number $t$, set $x = \dfrac{t - 1}{t + 1}$. Cross multiplying, $x(t + 1) = t - 1$, or $xt + x = t - 1$. Thus $x + 1 = t - xt$, i.e., $x + 1 = t(1 - x)$. Since $x \neq 1$ (or else t - 1 = t + 1, which is impossible), then $t = \dfrac{1 + x}{1 - x}$. The expression I have for $x$ is obtained by solving the rational equation $t = \dfrac{1-x}{1+x}$ for $x$.

To show that $-1 < x < 1$, observe that $t - 1 < t + 1$ and $-(t - 1) = -t + 1 < t + 1$ (since $t$ is positive). Therefore $-(t+1) < t - 1 < t + 1$; dividing by $t + 1$, we obtain $-1 < x < 1$, as desired.
 

Related to Using Standard Taylor Series to build other Taylor Series

1. What is the purpose of using Standard Taylor Series to build other Taylor Series?

The purpose of using Standard Taylor Series is to approximate a function with a polynomial expression. This can be useful in various mathematical and scientific applications, such as finding derivatives, integrals, and solving differential equations.

2. How do you use Standard Taylor Series to build other Taylor Series?

To use Standard Taylor Series to build other Taylor Series, one must first determine the center point of the series and the number of terms to include. Then, the coefficients of the polynomial expression can be calculated using the formula for the nth derivative of the function at the center point.

3. Can Standard Taylor Series be used for any function?

No, Standard Taylor Series can only be used for functions that are infinitely differentiable at the center point. This means that the function must have derivatives of all orders at that point.

4. What is the significance of including more terms in a Taylor Series?

Including more terms in a Taylor Series allows for a more accurate approximation of the function. The more terms that are included, the closer the polynomial expression will match the original function.

5. Are there any limitations to using Standard Taylor Series to build other Taylor Series?

Yes, there are limitations. Standard Taylor Series can only be used to approximate functions in a specific interval around the center point. This means that the accuracy of the approximation may decrease as the distance from the center point increases.

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