Using
expand is a Taylor series about z=2.
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Do I just find all the derivatives or something? Perhaps I need to use a geometric series here? Partial fractions? How do I start?
EDIT: It also mentions I need the first four non-zero terms of the series.
Assume you can write as a Taylor polynomial.
So .
By letting we can see that .
Differentiate both sides
.
By letting we can see that .
Differentiate both sides
.
By letting we can see that .
Differentiate both sides
.
By letting we can see that .
Differentiate both sides
.
By letting we can see that .
I think you can see that the series is taking the form
.
Now you need to check the values for which this series will converge.
Since this is a geometric series with common ratio , it only converges for
.
multiply top and bottom by 1/3 and re-express the function as 1/3*(-1/(2-z)/3) now substitute (2-z)/3 into the geometric expansion and the rest is algebra! God I wish I could express myself better symbolically. Much less messy than ProveIt's approach!