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mahmoud2011 · 2012-03-21T19:59:12+00:00

Is this inequality true ?? 0\leq \sum_{k=0}^{n} \frac{1}{(k+1)^2 (n-k+1)} \leq \frac{1}{\sqrt{n+1}} for all natural numbers n Is it true ?? Thanks

Thread starter mahmoud2011
Start date Mar 21, 2012
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Inequality

In summary, there are various methods for proving the truth of an inequality, such as using algebraic manipulation or mathematical induction. Inequalities cannot be solved in the same way as equations, but they can be simplified and manipulated. To determine the truth of an inequality graphically, one can plot the inequality on a graph and see if the shaded region aligns with the given inequality. Inequalities can be used with all types of numbers, but the methods for solving and manipulating may differ. Some common rules for solving inequalities include reversing the direction of the inequality when multiplying or dividing by a negative number, combining like terms, and using the distributive property. It is important to also consider any restrictions or conditions given in the problem.

Mar 21, 2012

mahmoud2011

Is this inequality true ??

[itex]0\leq \sum_{k=0}^{n} \frac{1}{(k+1)^2 (n-k+1)} \leq \frac{1}{\sqrt{n+1}} [/itex] for all natural numbers n

Is it true ??

Thanks

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Mar 21, 2012

disregardthat

Science Advisor

Can you write [tex]\frac{1}{(k+1)^2(n-k+1)}[/tex] differently? You want something on the form [tex]\frac{a}{k+1} + \frac{bk+c}{(k+1)^2} + \frac{d}{n-k+1}[/tex] .

Mar 23, 2012

mahmoud2011

It really work thanks .

Last edited: Mar 23, 2012

1. Is there a way to prove if an inequality is true or not?

Yes, there are various methods for proving the truth of an inequality. One common method is using algebraic manipulation to transform the inequality into a simpler form that is easier to evaluate. Another method is using mathematical induction to show that the inequality holds for all values within a given range.

2. Can inequalities be solved like equations?

No, inequalities cannot be solved in the same way that equations can. Inequalities involve a range of values rather than a single value, so the solution is a set of values that satisfy the inequality. Inequalities can be simplified and manipulated, but they cannot be solved in the traditional sense.

3. How can I determine the truth of an inequality graphically?

To determine the truth of an inequality graphically, you can plot the inequality on a graph and see if the shaded region aligns with the given inequality. If the shaded region does not align, then the inequality is not true. If the shaded region does align, then the inequality is true.

4. Can all types of numbers be used in inequalities?

Yes, inequalities can be used with all types of numbers, including integers, fractions, decimals, and even irrational numbers. However, the methods for solving and manipulating inequalities may differ depending on the type of numbers involved.

5. Are there any rules for solving inequalities?

Yes, there are several rules for solving inequalities. Some of the common rules include: when multiplying or dividing by a negative number, the direction of the inequality symbol must be reversed; combining like terms on both sides of the inequality; and using the distributive property to simplify expressions. It is important to also pay attention to any restrictions or conditions given in the problem.