# An Inequality by Uncommon Induction

Prove that for every \(n\gt 1\),

\(\displaystyle 1+\frac{1}{2^2}+\frac{1}{3^2}+\ldots +\frac{1}{n^2}\gt \frac{3n}{2n+1}. \)

|Contact| |Front page| |Contents| |Algebra| |Store|

Copyright © 1996-2017 Alexander Bogomolny

Prove that for every \(n\gt 1\),

\(\displaystyle 1+\frac{1}{2^2}+\frac{1}{3^2}+\ldots +\frac{1}{n^2}\gt \frac{3n}{2n+1}. \)

The first idea that comes to mind is that the method of mathematical induction ought to be of use for the proof. This is indeed so, but not without a workaround. For \(n=1\), the two expressions are equal: \(\displaystyle 1=\frac{3\cdot 1}{2\cdot 1+1}\), and this is why \(n=1\) is excluded. From then on, the two sides grow. The left-hand side grows by \(\displaystyle \frac{1}{n^2}\), the right-hand side grows by

\(\displaystyle \frac{3n}{2n+1} - \frac{3(n-1)}{2(n-1)+1} = \frac{3}{4n^{2}-1}. \)

Now, it is easy to verify that, for \(n\gt 1\), \(\displaystyle \frac{1}{n^2}\gt \frac{3}{4n^{2}-1}\). This exactly means that the left-hand side grows faster than the right-hand side which, thus, proves the inequality.

The two sides monotone increasing as \(n\rightarrow\infty\); the left-hand side is known as Euler series, with the famous value:

\(\displaystyle \sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi ^2}{6}\approx 1.645. \)

This is certainly greater than the limit \(\displaystyle\frac{3}{2}\) of the right-hand side. In itself, though, this is not yet sufficient to prove the inequality for all \(n\gt 1\)!

Jack D'Aurizio came up with another solution. He starts with

\(\displaystyle \sum_{k=2}^{n}\frac{1}{k^2} \lt \sum_{k=2}^{n}\frac{1}{k^2-1/4} = 2\sum_{k=2}^{n}\bigg(\frac{1}{2k-1} - \frac{1}{2k+1}\bigg) = 2\bigg(\frac{1}{3} - \frac{1}{2n+1}\bigg), \)

which holds for \(n\ge 2\). By adding \(1\) to both sides we get:

\(\displaystyle \sum_{k=1}^{n}\frac{1}{k^2} > \frac{10n-1}{6n+3}, \)

which is stronger than the inequality we set out to prove, because \(\displaystyle\frac{10n-1}{6n+3}\gt \frac{3n}{2n+1}\), for \(n\gt 1.\)

If we apply the "telescoping estimation" technique later, we get even stronger inequalities. For example, starting from

\(\displaystyle \sum_{k=3}^{n}\frac{1}{k^2} \lt \sum_{k=3}^{n}\frac{1}{k^2-1/4} = 2\sum_{k=3}^{n}\bigg(\frac{1}{2k-1} - \frac{1}{2k+1}\bigg) = 2\bigg(\frac{1}{5} - \frac{1}{2n+1}\bigg) \)

and by adding \(\displaystyle\frac{5}{4}\) to both sides, we get

\(\displaystyle \sum_{k=1}^{n} \frac{1}{k^2} \gt \frac{66n-7}{20(2n+1)}, \)

which also holds for \(n\gt 1.\) Starting with \(k=4\) and adding \(\displaystyle\frac{5}{4}\) gives

\(\displaystyle \sum_{k=1}^{n} \frac{1}{k^2} \gt \frac{830 n - 89}{252 (2n+1)}, \)

### Reference

- R. Honsberger,
*More Mathematical Morsels*, MAA, New Math Library, 1991, 33-35

- An Inequality for Grade 8
- An Extension of the AM-GM Inequality
- Schur's Inequality
- Newton's and Maclaurin's Inequalities
- Rearrangement Inequality
- Chebyshev Inequality
- Jensen's Inequality
- Muirhead's Inequality
- Bergström's inequality
- Radon's Inequality and Applications
- Jordan and Kober Inequalities, PWW
- A Mathematical Rabbit out of an Algebraic Hat
- An Inequality With an Infinite Series
- An Inequality: 1/2 * 3/4 * 5/6 * ... * 99/100 less than 1/10
- A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
- An Inequality: Easier to prove a subtler inequality
- Inequality with Logarithms
- An inequality: 1 + 1/4 + 1/9 + ... less than 2
- Inequality with Harmonic Differences
- An Inequality by Uncommon Induction
- Hlawka's Inequality
- An Inequality in Determinants
- Application of Cauchy-Schwarz Inequality
- An Inequality from Tibet
- An Inequality with Constraint
- An Inequality from Morocco
- An Inequality for Mixed Means
- An Inequality in Integers
- An Inequality in Integers II
- An Inequality in Integers III
- An Inequality with Exponents
- Exponential Inequalities for Means
- A Simple Inequality in Three Variables
- An Asymmetric Inequality
- Linear Algebra Tools for Proving Inequalities
- An Inequality with a Generic Proof
- A Generalization of an Inequality from a Romanian Olympiad
- Area Inequality in Trapezoid
- Improving an Inequality
- RomanoNorwegian Inequality
- Inequality with Nested Radicals II
- Inequality with Powers And Radicals
- Inequality with Two Minima
- Simple Inequality with Many Faces And Variables
- An Inequality with Determinants
- An Inequality with Determinants II
- An Inequality with Determinants III
- An Inequality with Determinants IV
- An Inequality with Determinants V
- An Inequality with Determinants VI
- An Inequality with Determinants VII
- An Inequality in Reciprocals
- An Inequality in Reciprocals II
- An Inequality in Reciprocals III
- Monthly Problem 11199
- A Problem from the Danubius Contest 2016
- A Problem from the Danubius-XI Contest
- An Inequality with Integrals and Rearrangement
- An Inequality with Cot, Cos, and Sin
- A Trigonometric Inequality from the RMM
- An Inequality with Finite Sums
- Hung Viet's Inequality
- Hung Viet's Inequality II
- Hung Viet's Inequality III
- Inequality by Calculus
- Dorin Marghidanu's Calculus Lemma
- An Area Inequality
- A 4-variable Inequality from the RMM
- An Inequality from RMM with Powers of 2
- A Cycling Inequality with Integrals
- A Cycling Inequality with Integrals II
- An Inequality with Absolute Values
- An Inequality from RMM with a Generic 5
- An Elementary Inequality by Non-elementary Means
- Inequality in Quadrilateral
- Marian Dinca's Refinement of Nesbitt's Inequality
- An Inequality in Cyclic Quadrilateral
- An Inequality in Cyclic Quadrilateral II
- An Inequality in Cyclic Quadrilateral III
- An Inequality in Cyclic Quadrilateral IV
- Inequality with Three Linear Constraints
- Inequality with Three Numbers, Not All Zero
- An Easy Inequality with Three Integrals
- Divide And Conquer in Cyclic Sums
- Wu's Inequality
- A Cyclic Inequality in Three Variables
- Dorin Marghidanu's Inequality in Complex Plane
- Dorin Marghidanu's Inequality in Integer Variables
- Dorin Marghidanu's Inequality in Many Variables
- Dorin Marghidanu's Inequality in Many Variables Plus Two More
- Dorin Marghidanu's Inequality with Radicals
- Dorin Marghidanu's Light Elegance in Four Variables
- Dorin Marghidanu's Spanish Problem
- Two-Sided Inequality - One Provenance
- An Inequality with Factorial
- Wonderful Inequality on Unit Circle
- Quadratic Function for Solving Inequalities
- An Inequality Where One Term Is More Equal Than Others
- An Inequality and Its Modifications
- Complicated Constraint - Simple Inequality
- Distance Inequality
- Two Products: Constraint and Inequality
- The power of substitution II: proving an inequality with three variables
- Algebraic-Geometric Inequality
- One Inequality - Two Domains
- Radicals, Radicals, And More Radicals in an Inequality
- An Inequality in Triangle and In General
- Cyclic Inequality with Square Roots
- Dan Sitaru's Cyclic Inequality In Many Variables
- An Inequality on Circumscribed Quadrilateral
- An Inequality with Fractions
- An Inequality with Complex Numbers of Unit Length
- An Inequality with Complex Numbers of Unit Length II
- Le Khanh Sy's Problem
- An Inequality Not in Triangle
- An Acyclic Inequality in Three Variables
- An Inequality with Areas, Norms, and Complex Numbers
- Darij Grinberg's Inequality In Three Variables
- Small Change Makes Big Difference
- Inequality with Two Variables? Think Again
- 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
- Twin Inequalities in Four Variables: Twin 1
- Twin Inequalities in Four Variables: Twin 2
- 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
- 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
- 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
- Dan Sitaru's Exercise with Pi and Ln
- Problem 4165 from Crux Mathematicorum

|Contact| |Front page| |Contents| |Algebra| |Store|

Copyright © 1996-2017 Alexander Bogomolny

62075070 |