# Induction to Prove Series Inequality

• tajmorton
In summary, the conversation discusses using induction to prove that the sum of a series is less than or equal to a given value. The approach involves showing that the sum is less than or equal to a value for n+1 by using the inductive hypothesis and then proving that the first inequality is true. The conversation also includes a helpful tip for simplifying the equation.
tajmorton

## Homework Statement

Show $$\sum_{i=1}^n \frac{1}{i^2}$$ $$\leq$$ $$2 - \frac{1}{n}$$ with induction on n.

I'm pretty rusty on induction (not that I was very good at it to being with), so I mostly wanted to know if I'm on the right track, and if this is a way towards a valid proof.

## The Attempt at a Solution

Base Step (n=1): 1 $$\leq$$ 2 - 1/1
(works)

Inductive Hypothesis:
$$\sum_{i=1}^n \frac{1}{i^2} = S_{j}$$
$$S_{j} \leq 2 - \frac{1}{j}$$ for all j = 0, 1, ... n

Inductive Step:
$$S_{n+1} = S_{n} + \frac{1}{{(n+1)}^2}$$

Using inductive assumption:
$$S_{n} + \frac{1}{{n+1}^2} \leq 2 - \frac{1}{n} + \frac{1}{{(n+1)}^2}$$

My plan is to show that
$$2 - \frac{1}{n} + \frac{1}{{(n+1)}^2} \leq 2 - \frac{1}{n+1}$$,
since by the inductive hypothesis we have
$$S_{n} \leq 2 - \frac{1}{n}$$.

So, if I can show that the first inequality is true, then I think the following should be true:
$$S_{n} \leq 2 - \frac{1}{n} + \frac{1}{{(n+1)}^2} \leq 2 - \frac{1}{n+1}$$

Would that be a valid proof?

Apologies about the poor TeX, I tried to fix some of the problems I saw, but they never updated in my browser (cache?).
Any pointers would be appreciated... Thanks!
- Taj

Sounds good, and perhaps this would help;

$$\frac{1}{n} - \frac{1}{(n+1)^2} = \frac{1}{n+1} \left( \frac{n+1}{n} - \frac{1}{n + 1} \right) = \frac{1}{n+1} \left( 1 + \frac{1}{n} - \frac{1}{n + 1} \right)$$

## 1. What is induction to prove series inequality?

Induction to prove series inequality is a mathematical method used to prove that a certain inequality is true for all natural numbers. It involves breaking down the inequality into smaller cases and proving that it holds for each case, thus establishing that it holds for all values.

## 2. Why is induction used to prove series inequality?

Induction is used to prove series inequality because it provides a systematic and rigorous way of proving that a statement is true for all natural numbers. It allows us to generalize from a few specific cases to an infinite number of cases.

## 3. What are the steps involved in using induction to prove series inequality?

The steps involved in using induction to prove series inequality are: 1. Prove the base case - show that the statement holds for the first value of the natural numbers2. Assume the statement holds for some value of the natural numbers3. Use this assumption to prove that the statement holds for the next value of the natural numbers4. Repeat this process until the statement is proven for all natural numbers.

## 4. Can induction be used to prove any inequality?

No, induction can only be used to prove inequalities that hold for all natural numbers. It cannot be used for inequalities that only hold for specific values or inequalities involving real numbers.

## 5. Are there any limitations to using induction to prove series inequality?

One limitation of using induction to prove series inequality is that it can only be used to prove statements for natural numbers. It cannot be used for other sets of numbers such as integers or real numbers. Additionally, for more complex inequalities, the induction process may become more difficult and time-consuming.

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