Find the sum of the series `sum_(r=11)^(99)(1/(rsqrt(r+1)+(r+1)sqrtr))`

Explanation

To find the sum of the series

\[ S = \sum_{r=11}^{99} \frac{1}{r\sqrt{r+1} + (r+1)\sqrt{r}}, \]

we will simplify the expression step by step.

### Step 1: Simplify the expression
We start with the term in the sum:

\[ \frac{1}{r\sqrt{r+1} + (r+1)\sqrt{r}}. \]

We can factor out \(\sqrt{r}\sqrt{r+1}\) from the denominator:

\[ = \frac{1}{\sqrt{r}\sqrt{r+1} \left(\frac{r}{\sqrt{r}} + \frac{r+1}{\sqrt{r+1}}\right)}. \]

### Step 2: Rationalize the denominator
Next, we rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{r+1} - \sqrt{r}\):

\[ = \frac{\sqrt{r+1} - \sqrt{r}}{(r\sqrt{r+1} + (r+1)\sqrt{r})(\sqrt{r+1} - \sqrt{r})}. \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:

\[ (r\sqrt{r+1} + (r+1)\sqrt{r})(\sqrt{r+1} - \sqrt{r}) = (r+1) - r = 1. \]

Thus, we have:

\[ = \sqrt{r+1} - \sqrt{r}. \]

### Step 4: Rewrite the sum
Now, we can rewrite the sum \(S\) as:

\[ S = \sum_{r=11}^{99} (\sqrt{r+1} - \sqrt{r}). \]

### Step 5: Recognize the telescoping series
This is a telescoping series. When we expand it, we see that most terms will cancel:

\[ S = (\sqrt{12} - \sqrt{11}) + (\sqrt{13} - \sqrt{12}) + \ldots + (\sqrt{100} - \sqrt{99}). \]

### Step 6: Evaluate the sum
After cancellation, we are left with:

\[ S = \sqrt{100} - \sqrt{11} = 10 - \sqrt{11}. \]

### Final Answer
Thus, the sum of the series is:

\[ S = 10 - \sqrt{11}. \]

---