  
1
√
2
+
1
√
3
+
...
+
1
√
10
>
2
 

Let's analyze the inequality:

\[
\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{10}} > 2
\]

To evaluate this sum, we'll first calculate the value of each term on the left-hand side:

\[
\frac{1}{\sqrt{2}} \approx 0.7071, \quad \frac{1}{\sqrt{3}} \approx 0.5774, \quad \frac{1}{\sqrt{4}} = 0.5
\]

Continuing in this manner:

\[
\frac{1}{\sqrt{5}} \approx 0.4472, \quad \frac{1}{\sqrt{6}} \approx 0.4082
\]
\[
\frac{1}{\sqrt{7}} \approx 0.3770, \quad \frac{1}{\sqrt{8}} \approx 0.3536
\]
\[
\frac{1}{\sqrt{9}} = 0.3333, \quad \frac{1}{\sqrt{10}} \approx 0.3162
\]

Adding these together:

\[
0.7071 + 0.5774 + 0.5 + 0.4472 + 0.4082 + 0.3770 + 0.3536 + 0.3333 + 0.3162 \approx 4.02
\]

Since \( 4.02 > 2 \), the inequality is indeed true.

### Do you want details or have any questions? 

#### Related Questions:
1. How can we approximate sums of series involving square roots?
2. Can inequalities involving sums of fractions always be compared to an integer?
3. What is the significance of the series convergence in this context?
4. How does this series compare to a harmonic series?
5. How can we generalize this inequality to a larger sum?

#### Tip:
When dealing with sums involving square roots, calculating the first few terms often helps determine the overall behavior of the sum.

1 / √2 + 1 / √3 + ... + 1 / √10 > 2

Explore the inequality 1/√2 + 1/√3 + ... + 1/√10 > 2 with detailed step-by-step calculations. Learn how to handle sums of square roots and verify the inequality with accurate summation. Perfect for students in Grades 10-12, this guide covers inequalities, series summation, and square root approximations.

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