Suppose that we want to prove that (1)/(2) ·(3)/(4) ⋯(2 n-1)/(2 n)<(1)/(√(3 n)) for all positive

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Suppose that we want to prove that (1)/(2) ·(3)/(4) ⋯(2...

Suppose that we want to prove that $$ \frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2 n-1}{2 n}<\frac{1}{\sqrt{3 n}} $$ for all positive integers $n$. a) Show that if we try to prove this inequality using mathematical induction, the basis step works, but the inductive step fails. b) Show that mathematical induction can be used to prove the stronger inequality $$ \frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2 n-1}{2 n}<\frac{1}{\sqrt{3 n+1}} $$ for all integers greater than 1 , which, together with a verification for the case where $n=1$, establishes the weaker inequality we originally tried to prove using mathematical induction.

Suppose that we want to prove that
$$
\frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2 n-1}{2 n}<\frac{1}{\sqrt{3 n}}
$$
for all positive integers $n$.
a) Show that if we try to prove this inequality using mathematical induction, the basis step works, but the inductive step fails.
b) Show that mathematical induction can be used to prove the stronger inequality
$$
\frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2 n-1}{2 n}<\frac{1}{\sqrt{3 n+1}}
$$
for all integers greater than 1 , which, together with a verification for the case where $n=1$, establishes the weaker inequality we originally tried to prove using mathematical induction.

Key Concepts

Inequalities Involving Product Sequences

Inequalities involving product sequences, such as products of rational numbers, require careful handling because successive multiplication can amplify errors or bounds. Proving these inequalities often involves establishing tight control over each factor and sometimes transforming the product into a more manageable form. This may involve comparing the product to a simpler function whose behavior is well understood, thus facilitating the application of induction or other analytical methods.

Strengthening an Induction Hypothesis

Occasionally, the induction hypothesis as stated is too weak to make the inductive step work. In such cases, the proof can be restructured to prove a stronger inequality that implies the original claim. By demonstrating the stronger statement holds for all natural numbers, one can also derive the original, weaker inequality as a consequence. This method ensures that the inductive step is achievable and that the final result is valid.

Mathematical Induction

Mathematical induction is a fundamental proof technique used to establish that a statement holds for all natural numbers. It involves two main steps: verifying that the statement is true for an initial case (basis step), and then proving that if the statement holds for an arbitrary case n (the induction hypothesis), it will also hold for the next case, n+1 (inductive step). This technique ensures that the property is propagated through all natural numbers.

Inductive Step in Proofs

The inductive step is critical in using mathematical induction. It involves assuming the claim holds for a particular integer n and then showing that this assumption logically implies the claim for n+1. However, sometimes directly establishing the inductive step with the original statement is not straightforward, which may cause the approach to fail unless additional insights or modifications are introduced.

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