- #1

lyd123

- 13

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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.