Use mathematical induction to prove the inequalities in...

Use mathematical induction to prove the inequalities in Exercises $18-30 .$ Let $P(n)$ be the statement that $$ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^{2}}<2-\frac{1}{n} $$ where $n$ is an integer greater than 1 a) What is the statement $P(2) ?$ b) Show that $P(2)$ is true, completing the basis step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1 . c) What is the inductive hypothesis of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1$?$ d) What do you need to prove in the inductive step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1$?$ e) Complete the inductive step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than $1 .$ f) Explain why these steps show that this inequality is true whenever $n$ is an integer greater than 1 .

Use mathematical induction to prove the inequalities in Exercises $18-30 .$
Let $P(n)$ be the statement that
$$
1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^{2}}<2-\frac{1}{n}
$$
where $n$ is an integer greater than 1
a) What is the statement $P(2) ?$
b) Show that $P(2)$ is true, completing the basis step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1 .
c) What is the inductive hypothesis of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1$?$
d) What do you need to prove in the inductive step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than 1$?$
e) Complete the inductive step of a proof by mathematical induction that $P(n)$ is true for all integers $n$ greater than $1 .$
f) Explain why these steps show that this inequality is true whenever $n$ is an integer greater than 1 .

Key Concepts

Series Inequalities

Series inequalities involve proving that a finite or infinite sum meets certain inequality conditions. In the context of induction proofs, it often requires establishing a relationship between the sum of a series up to a certain term and a function of that term, and then using induction to prove that this relationship holds for all terms in the series.

Inductive Step

The inductive step is the second part of a proof by mathematical induction. In this step, one shows that under the assumption that the statement holds for an arbitrary integer k (the inductive hypothesis), it must also hold for the next integer, k+1. Successfully completing this step proves the viability of the inductive process and leads to the conclusion that the statement is true for all integers in the domain.

Inductive Hypothesis

The inductive hypothesis is the assumption made during an inductive proof that the statement holds true for some arbitrary integer k. This hypothesis is not proved but assumed to be true in order to demonstrate that if the statement holds for k, then it logically follows that the statement must also hold for k+1.

Base Case

The base case is the first step in a proof by mathematical induction. It involves verifying that the statement is true for the initial value of the integer variable (often n=1 or another specified starting point). This step is critical because it establishes the start of the inductive chain that supports the entire proof.

Mathematical Induction

Mathematical induction is a proof technique used to prove statements or propositions that are formulated for all integers greater than or equal to a certain base value. It involves two main steps: proving the statement for an initial integer (known as the base case) and then proving that if the statement holds for an arbitrary integer k, it must also hold for k+1 (the inductive step), thereby establishing the truth of the statement for all subsequent integers.

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