Show 1/(n+1)+1/(n+2)+1/(n+3)+1/2n>3/5, for n>2. Use mathematical induction.
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I think the statement is correct. To prove it by induction, write the nth and (n+1)st sums: ; . Then express as where is some expression of n, i.e., . If you show that , you are done.
Note that has three terms, and two of them are at least as big as half of the third.