An Inequality:
(1 + 1^{-3})(1 + 2^{-3})(1 + 3^{-3})...(1 + n^{-3}) < 3
Prove the following inequality for all integer n greater than 0:
(1) | (1 + 1^{-3})(1 + 2^{-3})(1 + 3^{-3})...(1 + n^{-3}) < 3 |
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Copyright © 1996-2017 Alexander Bogomolny
Mathematical induction is a reasonable method to apply to proving (1) "for all n". As is the case with another example, there does not appear to be an obvious way to make induction work for (1).
(1) | (1 + 1^{-3})(1 + 2^{-3})(1 + 3^{-3})...(1 + n^{-3}) < 3 |
However, a strengthened inequality
(2) | (1 + 1^{-3})(1 + 2^{-3})(1 + 3^{-3})...(1 + n^{-3}) < 3 - 1/n |
is easily amenable to an inductive argument. Denote the left-hand side of (2) as A(n). The verification of (2) for
For n = k+1, we have
A(k+1) | = A(k)(1 + (k+1)^{-3}) |
< (3 - 1/k)(1 + 1/(k+1)^{3}) | |
= 3 - 1/k + 3/(k+1)^{3} - 1/[k(k+1)^{3}] | |
= 3 - ((k+1)^{3} - 3k + 1)/[k(k+1)^{3}]. |
Our goal of proving (2) for n = k+1 will be achieved provided
The latter simplifies to
and further to
(3) | k^{2} - k + 2 > 0. |
Since P(x) = x^{2} - x + 2 has no real roots and
We thus obtain an example of two problems - one weaker, the other stronger - of which the weaker problem is more difficult to prove than the stronger one. This fits into the observation that more general problem are often easier than their specific instances.
References
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- Dorin Marghidanu's Calculus Lemma
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- Inequality in Quadrilateral
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- Inequality with Three Linear Constraints
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- A Cyclic Inequality in Three Variables
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- Wonderful Inequality on Unit Circle
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- Distance Inequality
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- Radicals, Radicals, And More Radicals in an Inequality
- An Inequality in Triangle and In General
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- An Inequality Not in Triangle
- An Acyclic Inequality in Three Variables
- An Inequality with Areas, Norms, and Complex Numbers
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- Small Change Makes Big Difference
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- A Problem From a Mongolian Olympiad for Grade 11
- Sitaru--Schweitzer Inequality
- An Inequality with Cyclic Sums And Products
- Problem 1 From the 2016 Pan-African Math Olympiad
- An Inequality with Integrals and Radicals
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- Simple Inequality with a Variety of Solutions
- A Partly Cyclic Inequality in Four Variables
- Dan Sitaru's Inequality by Induction
- An Inequality in Three (Or Is It Two) Variables
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- An Inequality in Fractions with Absolute Values
- Inequalities with Double And Triple Integrals
- An Old Inequality
- Dan Sitaru's Amazing, Never Ending Inequality
- Leo Giugiuc's Exercise
- Another Inequality with Logarithms, But Not Really
- A Cyclic Inequality of Degree Four
- An Inequality Solved by Changing Appearances
- Distances to Three Points on a Circle
- An Inequality with Powers And Logarithm
- Four Integrals in One Inequality
- Same Integral, Three Intervals
- Dorin Marghidanu's Inequality with Generalization
- Dan Sitaru's Inequality with Three Related Integrals and Derivatives
- An Inequality in Two Or More Variables
- An Inequality in Two Or More Variables II
- A Not Quite Cyclic Inequality
- Dan Sitaru's Inequality: From Three Variables to Many in Two Ways
- An Inequality with Sines But Not in a Triangle
- An Inequality with Angles and Integers
- Sladjan Stankovik's Inequality In Four Variables
- An Inequality with Two Pairs of Triplets
- A Refinement of Turkevich's Inequality
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- Three Variables, Three Constraints, Two Inequalities (Only One to Prove) - by Leo Giugiuc
- An inequality in 2+2 variables from SSMA magazine
- Kunihiko Chikaya's Inequality with Parameter
- Dorin Marghidanu's Permuted Inequality
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Copyright © 1996-2017 Alexander Bogomolny
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