Sum of the series `P=(1)/(2sqrt(1)+sqrt(2))+(1)/(3sqrt(2)+2sqrt(3))+.........+(1)/(100sqrt(99)+99sqrt(100))` is

Options

A. `(1)/(10)`
B. `(3)/(10)`
C. `(9)/(10)`
D. `(1)/(2)`

Explanation

To solve the series

\[ P = \sum_{n=2}^{100} \frac{1}{n\sqrt{n-1} + (n-1)\sqrt{n}} \]

we will simplify each term in the series.

### Step 1: Simplify the general term

Consider the general term:

\[ \frac{1}{n\sqrt{n-1} + (n-1)\sqrt{n}} \]

We can multiply and divide by the conjugate:

\[ \frac{1}{n\sqrt{n-1} + (n-1)\sqrt{n}} \cdot \frac{n\sqrt{n-1} - (n-1)\sqrt{n}}{n\sqrt{n-1} - (n-1)\sqrt{n}} = \frac{n\sqrt{n-1} - (n-1)\sqrt{n}}{(n\sqrt{n-1})^2 - ((n-1)\sqrt{n})^2} \]

### Step 2: Simplify the denominator

The denominator simplifies as follows:

\[ (n\sqrt{n-1})^2 - ((n-1)\sqrt{n})^2 = n^2(n-1) - (n-1)^2n = n(n-1)(n - (n-1)) = n(n-1) \]

Thus, we have:

\[ \frac{n\sqrt{n-1} - (n-1)\sqrt{n}}{n(n-1)} \]

### Step 3: Rewrite the series

Now, we can rewrite the series \(P\):

\[ P = \sum_{n=2}^{100} \frac{n\sqrt{n-1} - (n-1)\sqrt{n}}{n(n-1)} \]

### Step 4: Split the terms

This can be split into two separate sums:

\[ P = \sum_{n=2}^{100} \left( \frac{\sqrt{n-1}}{n-1} - \frac{\sqrt{n}}{n} \right) \]

### Step 5: Recognize the telescoping nature

Notice that this series is telescoping. The first part of the sum will cancel with the second part of the next term:

\[ P = \left( \sqrt{1} - \sqrt{2} \right) + \left( \sqrt{2} - \sqrt{3} \right) + \ldots + \left( \sqrt{99} - \sqrt{100} \right) \]

### Step 6: Evaluate the sum

When we sum these terms, we see that all intermediate terms cancel out, leaving us with:

\[ P = \sqrt{1} - \sqrt{100} = 1 - 10 = -9 \]

### Final Result

Thus, the sum of the series \(P\) is:

\[ \boxed{-9} \]