- #1

MathematicalPhysicist

Gold Member

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exists. show that the limit is less than 1 but not less than 1/2.

the first part of the question i did already, im not sure about the second part of the question if i did properly.

n(1/n+...+1/2n)>=an=1/n+...+1/2n>=1/n^2+...+1/2n^2

because the limit exists, for every n>N(e) (for every e>0) |a-an|<e

and thus, a-e<an<a+e

if we let e=1/2 then a+e<=1.5 and thus a<=1.

a-1/2>=1/n^2+...+1/2n^2-1/2>0

thus a>1/2.

is this correct?

thanks in advance.