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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- An Inequality in Determinants
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- Inequality with Powers And Radicals
- Inequality with Two Minima
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- Inequality with Three Linear Constraints
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- A Cyclic Inequality in Three Variables
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- An Inequality in Triangle and In General
- Cyclic Inequality with Square Roots
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- Small Change Makes Big Difference
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- Dan Sitaru's Inequality by Induction
- An Inequality in Three (Or Is It Two) Variables
- An Inequality in Four Weighted Variables
- 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
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- 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
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Copyright © 1996-2017 Alexander Bogomolny
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