- #1
MathematicalPhysicist
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prove that the limit of the sequence an=1/n+1/(n+1)+...+1/2n
exists. show that the limit is less than 1 but not less than 1/2.
the first part of the question i did already, I am 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.
exists. show that the limit is less than 1 but not less than 1/2.
the first part of the question i did already, I am 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.