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To analyze and solve the given mathematical expression, let's break it down step by step.

The expression is:
\[
\sum_{i=2}^n \frac{1}{i^2} \leq 1 + \sum_{i=2}^n \frac{1}{i(i-1)}
\]

### Step 1: Understanding the Left Side of the Inequality
The left side is a summation:
\[
\sum_{i=2}^n \frac{1}{i^2}
\]
This represents the sum of the reciprocals of the squares of integers from \(i=2\) to \(n\).

### Step 2: Analyzing the Right Side of the Inequality
The right side has two parts:
1. The constant 1.
2. The summation:
   \[
   \sum_{i=2}^n \frac{1}{i(i-1)}
   \]
   Let's simplify this summation term.

### Step 3: Simplifying the Summation on the Right Side
The term \(\frac{1}{i(i-1)}\) can be rewritten using partial fractions:
\[
\frac{1}{i(i-1)} = \frac{1}{i-1} - \frac{1}{i}
\]
So, the summation becomes:
\[
\sum_{i=2}^n \left(\frac{1}{i-1} - \frac{1}{i}\right)
\]
This is a telescoping series, where most terms cancel out:
\[
\left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{n-1} - \frac{1}{n}\right)
\]
After cancellation, we are left with:
\[
1 - \frac{1}{n}
\]

### Step 4: Combining the Terms on the Right Side
So, the right side of the inequality simplifies to:
\[
1 + \left(1 - \frac{1}{n}\right) = 2 - \frac{1}{n}
\]

### Step 5: Forming the Final Inequality
The inequality now looks like:
\[
\sum_{i=2}^n \frac{1}{i^2} \leq 2 - \frac{1}{n}
\]
This expresses that the sum of the reciprocals of squares from 2 to \(n\) is less than or equal to \(2 - \frac{1}{n}\).

Would you like further details or clarifications on any step?

### Relative Questions:
1. How can telescoping series be applied to different types of sums?
2. What is the significance of inequalities in summations?
3. Can the sum \(\sum_{i=2}^n \frac{1}{i^2}\) be bounded differently for large \(n\)?
4. How do partial fractions simplify complex rational expressions?
5. How does the value of \(n\) affect the accuracy of the inequality?

### Tip:
Understanding telescoping series is crucial for simplifying and evaluating complex summations. Look for patterns in the terms that cancel out in sequences.

Analyze and simplify the given mathematical inequality.

Explore the step-by-step solution to the inequality \(\sum_{i=2}^n \frac{1}{i^2} \leq 1 + \sum_{i=2}^n \frac{1}{i(i-1)}\). This problem covers advanced concepts in series, telescoping series, and partial fractions, making it suitable for high school and undergraduate students.

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