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kingwinner
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My exam is coming up, I have 2 questions on infinite series. Any help is appreciated!
Quesetion 1) http://www.geocities.com/asdfasdf23135/calexam1.JPG
For part a, I got:
g(x)= Sigma (n=0, infinity) [(-1)^n * x^(2n)]
For part b, I got:
x
∫ tan^-1 (t^2) dt = Sigma (k=0, infinity) [(-1)^k * x^(4k+3)] / [(2k+1) (4k+3)]
0
But the part b, they also ask for the radius of convergence, how can I find it? Should I apply the ratio test to this series expansion (colored in red) to find the radius of convergence? Is there any faster way?
Question 2) Suppose f(x)= x cos(x^2), find f^(4101) (0).
[f^(n) (0) is the "n"th derivative evaluated at 0]
Should I use the fact that "On its interval of convergence, a power series is the Taylor series of its sum" to do this question?
So is it true that the power series of x cos(x^2) is EQUAL to the Taylor series of x cos(x^2) for ALL real numbers x?
Is there any difference between power series and Taylor series?
Thanks for your help!
Quesetion 1) http://www.geocities.com/asdfasdf23135/calexam1.JPG
For part a, I got:
g(x)= Sigma (n=0, infinity) [(-1)^n * x^(2n)]
For part b, I got:
x
∫ tan^-1 (t^2) dt = Sigma (k=0, infinity) [(-1)^k * x^(4k+3)] / [(2k+1) (4k+3)]
0
But the part b, they also ask for the radius of convergence, how can I find it? Should I apply the ratio test to this series expansion (colored in red) to find the radius of convergence? Is there any faster way?
Question 2) Suppose f(x)= x cos(x^2), find f^(4101) (0).
[f^(n) (0) is the "n"th derivative evaluated at 0]
Should I use the fact that "On its interval of convergence, a power series is the Taylor series of its sum" to do this question?
So is it true that the power series of x cos(x^2) is EQUAL to the Taylor series of x cos(x^2) for ALL real numbers x?
Is there any difference between power series and Taylor series?
Thanks for your help!
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