- #1
KOO
- 19
- 0
Prove this inequality by induction for all nEN:
1/√1 + 1/√2 + 1/√3 + ... + 1/√n >= √n
1/√1 + 1/√2 + 1/√3 + ... + 1/√n >= √n
The purpose of using induction is to prove that the inequality 1/√n holds for all natural numbers. This method involves proving that the inequality holds for a base case (usually n = 1), and then showing that if it holds for a particular value of n, it also holds for the next value (n+1). By repeating this process, we can prove that the inequality holds for all natural numbers.
To prove the base case of the inequality, we substitute n = 1 into the inequality and show that it holds true. This can be done by simplifying the expression 1/√1 and showing that it is less than or equal to the right side of the inequality.
The inductive hypothesis is the assumption that the inequality holds for a particular value of n. In the proof of 1/√n, the inductive hypothesis is that 1/√n ≤ 1/√(n+1).
To prove the inequality for the next value (n+1), we use the inductive hypothesis to show that if the inequality holds for n, it also holds for n+1. This can be done by substituting n+1 into the inequality and then using the inductive hypothesis to simplify and show that it is still true.
Yes, induction can be used to prove many different types of inequalities. It is a powerful mathematical tool that can be applied to a variety of problems, not just the inequality 1/√n. However, it is important to note that induction is not always the most efficient or appropriate method for proving inequalities, and other techniques may be more suitable in certain cases.