The sum to infinite terms of the series `cot^(- 1)(2^2+1/2)+cot^(- 1)(2^3+1/(2^2))+cot^(- 1)(2^4+1/(2^3))+...` is

Answer

To find the sum to infinite terms of the series

\[ \cot^{-1}\left(2^2 + \frac{1}{2}\right) + \cot^{-1}\left(2^3 + \frac{1}{2^2}\right) + \cot^{-1}\left(2^4 + \frac{1}{2^3}\right) + \ldots \]

we can follow these steps:

### Step 1: Identify the general term of the series
The general term of the series can be expressed as:

\[ \cot^{-1}\left(2^r + \frac{1}{2^{r-1}}\right) \quad \text{for } r = 2, 3, 4, \ldots \]

### Step 2: Rewrite the general term
We can rewrite the term inside the cotangent inverse function:

\[ \cot^{-1}\left(2^r + \frac{1}{2^{r-1}}\right) = \cot^{-1}\left(\frac{2^{r+1} + 1}{2^r}\right) \]

### Step 3: Use the identity for cotangent inverse
Using the identity

\[ \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \]

we can express the general term as:

\[ \cot^{-1}\left(2^r + \frac{1}{2^{r-1}}\right) = \tan^{-1}\left(\frac{2^r}{2^{r+1} + 1}\right) \]

### Step 4: Find the sum of the series
Now we can express the sum of the series as:

\[ S = \sum_{r=2}^{\infty} \tan^{-1}\left(\frac{2^r}{2^{r+1} + 1}\right) \]

### Step 5: Use the telescoping nature of the series
Notice that:

\[ \tan^{-1}(a) - \tan^{-1}(b) = \tan^{-1}\left(\frac{a-b}{1+ab}\right) \]

This suggests that we can express the series in a telescoping form. We can rewrite:

\[ \tan^{-1}(2^{r+1}) - \tan^{-1}(2^r) \]

Thus, the series can be rewritten as:

\[ S = \left(\tan^{-1}(2^3) - \tan^{-1}(2^2)\right) + \left(\tan^{-1}(2^4) - \tan^{-1}(2^3)\right) + \ldots \]

### Step 6: Evaluate the limit as \( r \to \infty \)
As \( r \) approaches infinity, \( \tan^{-1}(2^r) \) approaches \( \frac{\pi}{2} \). Therefore, the sum becomes:

\[ S = \lim_{n \to \infty} \left(\tan^{-1}(2^{n+1}) - \tan^{-1}(2^2)\right) = \frac{\pi}{2} - \tan^{-1}(4) \]

### Step 7: Final expression
Using the identity \( \tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2} \):

\[ S = \frac{\pi}{2} - \tan^{-1}(4) = \cot^{-1}(4) \]

Thus, the sum to infinite terms of the series is:

\[ \boxed{\tan^{-1}(2)} \]