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Calculus: Find the power series representation and the radius of convergence of the following:?
Calculus: Find the power series representation and the radius of convergence of the following:
f(x) = 1 / ((1+x)^2)
Please show the work and give an explanation.
I will pick a best answer. Thank you!
Start with the geometric series 1/(1 - t) = sum(k=0 to infinity) t^k.
This converges when |t| < 1.
Let t = -x:
1/(1 + x) = sum(k=0 to infinity) (-1)^k x^k.
This converges when |-x| = |x| < 1.
Now differentiate both sides (this does not affect the interval of convergence):
-1/(1 + x)^2 = sum(k=0 to infinity) (-1)^k k x^(k-1).
==> 1/(1 + x)^2 = sum(k=0 to infinity) (-1)^(k+1) k x^(k-1).
(This converges for |x| < 1.)
I hope this helps!
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