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Question: problem 2 im just having a hard time relating the...

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Problem 2, I’m just having a hard time relating the question to part 1b). Any help would be appreciated.
1. This problem is concerned with the geometric series Σnooz (a) Let sk(x) = Σn=oz = 1 + x + x2 +z?+ +rk-1 +rk for all k E N and all E R Find an explicit formula for s(x). To do this, write out rs(x) explicitly, an find s(x) - zsk(z) prove that the series Σ000z converges pointwise for all z E (-1,1), and find an explicit expression for the sum f(x) of the series. (c) Prove that, for every b with 0 < b<1, the series converges uniformly to f on the interval [-b, b]. Work directly with the definition of uniform convergence. 2. Assume that it is valid to differentiate the series f(x)-Σ。oz term-by-term for lal < 1. Use this assumption, and the result of Problem 1(b), to find an explicit expression for the sum of the series Σ001 nr for c < 1.
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