What Techniques Ensure Convergence and Differentiation of Power Series?

[asif zaidi]

Hi:

I have 2 questions about Power Series. The 1st one is a h/w problem and the 2nd one is an example from a textbook which I am having difficulty to figure out.

Problem1:
Given a power series g(x) = sum(0 to inf) of x^i/i!. Determine interval of convergence and compute g'(x).

Problem1 Solution:
The 1st part I did - it is from -infinity to +infinity.
To compute g'(x) is it just ix^i-1 /i! ?
If not, please advise.


Problem 2:
This is from Thomas' Calculus textbook on pg 669 (I am paraphrasing)
They are saying that 1/1+x = Sum ( 0 to inf) of (-1)^i * x^i.
Given above we can see that log(1+x) = sum (0 to inf) of (-1)^i * x^i+1 / i+1

Problem 2 Questions
Now this 1st part of the question, I proved as follows and I would like to know if it is right.
Sum (0 to inf) of (-1^i)*x^i = Sum (1 to inf) of (-x^i). This is just a geometric series and it converges 1/1+x if |x| <1. Is this right?

For the second part: 1/1+x (given |x| <1): 1-x+x^2-x^3...
If I integrate each term above, I will get it to the series (-1^i)*x^i+1 / i+1. I am not sure about this approach. Would this be a right way of proving it.

 

[Dick]

It all looks fine to me. You should probably use more parentheses, things like x^i+1/i+1 are confusing. x^(i+1)/(i+1) is much nicer.