# How do I prove this? (summation problem)

• Gridvvk
In summary, the conversation discusses whether the inequality $$\sum_{i=1}^{n} x_i^2 > \frac{1}{n^2}(\sum_{i=1}^{n} x_i)^2$$ is true, and the use of the Cauchy-Schwartz inequality to prove it. The final conclusion is that the inequality is true, with the use of Cauchy-Schwartz.
Gridvvk
$$\sum_{i=1}^{n} x_i^2 > \frac{1}{n^2}(\sum_{i=1}^{n} x_i)^2$$

Note: each x_i is any observation (or statistic) it can be any real number and need not be constrained in anyway whatsoever, though you can take n > 1 and integer (i.e. there is at least two observations and the number of observations is discrete).

I'm not sure if this true or not, but part of my analysis to a particular problem assumed this was true, and I'm trying to prove it is indeed true (it seems to be case for any examples I come up with).

So far I came up with,
$$n^2 \sum_{i=1}^{n} x_i^2 > \sum_{i=1}^{n} x_i^2 + 2\sum_{i \neq j, i > j} x_ix_j$$
$$(n^2 - 1)\sum_{i=1}^{n}x_i^2 > 2\sum_{i \neq j,\: i > j} x_ix_j$$

and I'm not sure how to proceed from there.

Are you familiar with the Cauchy-Schwartz inequality?

micromass said:
Are you familiar with the Cauchy-Schwartz inequality?

Yes I am, but I'm not sure how to use it here. If I was interested in both a x_i and y_i then I would see how to use it here, but here I'm only looking at a x_i.

Gridvvk said:
Yes I am, but I'm not sure how to use it here. If I was interested in both a x_i and y_i then I would see how to use it here, but here I'm only looking at a x_i.

Maybe take all ##y_i = 1##?

1 person
micromass said:
Maybe take all ##y_i = 1##?

Hmm. Alright then by Cauchy-Schwartz I can say,

$$(\sum_{i=1}^{n} x_i \times 1)^2 \le (\sum_{i=1}^{n}x_i^2) (\sum_{i=1}^{n}1) = n\sum_{i=1}^{n}x_i^2 < n^2 \sum_{i=1}^{n}x_i^2$$

Which was what I wanted.

Thanks!

## What is a summation problem?

A summation problem is a mathematical problem that involves finding the sum of a series of numbers, usually represented by the Greek letter sigma (∑).

## How do I solve a summation problem?

To solve a summation problem, you need to follow these steps:

1. Identify the pattern of the numbers in the series.
2. Determine the starting and ending values of the series.
3. Write out the series in summation notation, using the pattern and the starting and ending values.
4. Perform any necessary operations, such as addition or multiplication, on the numbers in the series.
5. Simplify the expression to get the final answer.

## What is the formula for a summation problem?

The formula for a summation problem is:

i=1n (ai) = a1 + a2 + ... + an

where n is the number of terms in the series, ai is the i-th term in the series, and a1 is the first term in the series.

## What are some common types of summation problems?

Some common types of summation problems include:

• Arithmetic series: where the difference between consecutive terms is constant.
• Geometric series: where the ratio between consecutive terms is constant.
• Finite series: where the series has a fixed number of terms.
• Infinite series: where the series has an infinite number of terms.

## Can I use a calculator to solve a summation problem?

Yes, you can use a calculator to solve a summation problem. Many scientific calculators have a "Sigma" button that allows you to input the necessary values and get the sum of the series. However, it is important to understand the steps involved in solving a summation problem manually before relying on a calculator.

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