Tuesday, 3 October 2017

real analysis - Proof for power series

I used half my Sunday for trying to proof the following, but I couldn't find the answer. Can you help me?





Let f(x)=k=0ak(xx0)k a power series with radius of convergence 0. Show: f(xn)=0 for a sequence of points {xn} with xnx0,xnx0ak=0 kN.



Hint: Define for jN
f(j)(x):=k=0aj+k(xx0)k

and use induction to prove for all jN:
f(jn)(xn)=0 for all nN and aj=0.


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