Friday, December 1, 2017

sequences and series - How to prove sumlimitsinftyi=1fraci2i converges?





What would be the simplest way to prove that i=1i2i converges?


Answer



Another possible way : consider i=1ixi=xi=1ixi1=xddx(i=1xi)=xddx(x1x)=x(1x)2

Now, replace x by 12 to get 2.



The same procedure would apply to i=1iai=a(a1)2


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