Wednesday, January 23, 2019

sequences and series - Show that sum+inftyn=1fracn2n=2


Show that +n=1n2n=2. I have no idea to solve this problem. Anyone could help me?


Answer



Consider the following



11x=k0xk


The series converges for |x|<1


Differentiating both sides we have


1(1x)2=k1kxk1


x(1x)2=k1kxk


Now put x=12


2=k1k2k


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