Hello !!
Give the Taylor series about 0 for f and determine a range of validity for the series
Using the standard taylor series
Substituting
And taking 2 in the binomial series gives
Gives
as
Then
is valid for
Thanks Macca
I'm going to try a different, simpler, approach (tell me if you see anything wrong)Originally Posted by macca101
we have the taylor series:
and we have the series...
now let's solve for:
but let's say
then rewrite the problem:
and then let's say
now remembering the taylor series:
we see that must be between -1 and 1, let's figure out what needs to be between:
and solve from there...
~
The Taylor series was found (hopefully) in the first part of my question.Originally Posted by ThePerfectHacker
What I was and sill am struggling with is finding the interval which the series is valid.
Quick gets it to
I have a math cad file that allows me to plot a Taylor series against the original function see below. This indicates to me the series is valid for approximately
I hope I am making myself clear here I'm not saying quick is wrong (far be it from me) I just don't understand how to find the interval algebraically
You mean geomteric series.Originally Posted by macca101
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I assume you want an infinite series for a function,
Note the infinite geometric,
for
Take derivative of both sides,
Substitute
Thus,
Good you got that right.
You are trying to find for what values this series converges.
You are going to use the generalized ratio test, with
Thus,
Its limit as, is 1/6,
Therefore the reciprocal of the limit tells you the radius of convergence which is 6.
Thus,
Same as saying,