Help with integral/logarithm inequality 1/(n+1)<int_n^(n+1)1/t

Ratuiszt 2022-09-27 Answered
I have to prove the following inequality:
1 / ( n + 1 ) < n n + 1 1 / t d t < 1 / n
I thought it would be easier to attack this via integration, so I get:
1 / ( n + 1 ) < log ( n + 1 ) log ( n ) < 1 / n
At this point I tried to use induction, but the solution is still not clear to me.
Thanks a lot!
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Answers (1)

Farbwolkenw
Answered 2022-09-28 Author has 6 answers
Hint: Draw a picture. On the interval [ n , n + 1 ], our function 1 t is 1 n , and 1 n + 1
So the area under the curve 1 t , and above the t-axis, from t = n to t = n + 1, is less than the area of a rectangle with base 1 and height 1 n , and greater than the area of a rectangle with base 1 and height 1 n + 1
From the above geometric argument, it follows that 1 n + 1 < log ( n + 1 ) log n < 1 n
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