the value of (1)/(\sqrt(11)+\sqrt(9))+(1)/(\sqrt(3)+\sqrt(11))+........+(1)/(\sqrt(2025)+\sqrt(2023)) is

The given series is:

\[
\frac{1}{\sqrt{11} + \sqrt{9}} + \frac{1}{\sqrt{13} + \sqrt{11}} + \cdots + \frac{1}{\sqrt{2025} + \sqrt{2023}}
\]

To simplify each term, we use the identity:

\[
\frac{1}{\sqrt{a} + \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a - b}
\]

For each term in the series, we have \(a\) and \(b\) such that \(a = 2k+9\) and \(b = 2k+7\) where \(k\) is an integer starting from 1 up to some maximum value where \(a = 2025\).

Rewriting the terms using the identity:

\[
\frac{1}{\sqrt{2k+9} + \sqrt{2k+7}} = \frac{\sqrt{2k+9} - \sqrt{2k+7}}{(2k+9) - (2k+7)} = \frac{\sqrt{2k+9} - \sqrt{2k+7}}{2}
\]

Thus, the series becomes:

\[
\sum_{k=1}^{1008} \frac{\sqrt{2k+9} - \sqrt{2k+7}}{2}
\]

This is a telescoping series. The terms will cancel out except for the first negative and the last positive terms:

\[
\frac{1}{2} \left( (\sqrt{11} - \sqrt{9}) + (\sqrt{13} - \sqrt{11}) + \cdots + (\sqrt{2025} - \sqrt{2023}) \right)
\]

In a telescoping series, most terms cancel out, leaving:

\[
\frac{1}{2} \left( \sqrt{2025} - \sqrt{9} \right)
\]

Calculating the remaining terms:

\[
\sqrt{2025} = 45, \quad \sqrt{9} = 3
\]

So:

\[
\frac{1}{2} (45 - 3) = \frac{1}{2} \cdot 42 = 21
\]

Therefore, the value of the given series is:

\[
\boxed{21}
\]

Would you like more details or have any questions?

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Here are 8 related questions to expand on this information:
1. What is a telescoping series and how does it simplify the summation of series?
2. How can you identify if a series is telescoping?
3. Can the identity used for simplifying terms in this series be applied to other series?
4. What are some other examples of telescoping series in mathematics?
5. How does the convergence of telescoping series compare to other series?
6. What are the advantages of converting series to telescoping forms?
7. Can you demonstrate another example of a series that can be simplified using the telescoping method?
8. How do different types of series affect the approaches used in evaluating their sums?

Tip: Recognizing patterns in series, such as telescoping, can greatly simplify complex summation problems.

Explore how to simplify and calculate a telescoping series involving square roots. Learn the identity used for simplifying terms in series with detailed step-by-step solutions. Suitable for advanced high school students and those interested in advanced mathematics.

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