Prove that with be a possitive integer and
.
Solution. For statement is correct. Consider
. If some of the sums
() then we are done, since in this case, due to induction proposition, whole sum is
.
Now suppose all mentioned sums are negative and whole sum is non-positive. Then sum them up
.
Since is positive in neighbourhood of
and
, we may WLOG assume
(just consider the largest
with condition
). Thus
.
Using well known formulas for and
we obtain
and
.
It immediatelly implies
a contradiction.
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