Find the sum of the series:tan^-1(1/3)+tan^-1(2/9)+tan^-1(4/33)+....+tan^-1(2^(n-1)/1+2^(2n-1)

To find the sum of the series given by \( \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{2}{9}\right) + \tan^{-1}\left(\frac{4}{33}\right) + \ldots + \tan^{-1\left(\frac{2^{n-1}}{1 + 2^{2n-1}}\right) \), we can utilize properties of the arctangent function and some clever algebraic manipulation.

Understanding the Series

The series consists of terms of the form \( \tan^{-1}\left(\frac{2^{k-1}}{1 + 2^{2k-1}}\right) \) for \( k = 1 \) to \( n \). To simplify this, we can use the identity for the tangent of the sum of two angles:

Using the Tangent Addition Formula

Recall that:

• \( \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x + y}{1 - xy}\right) \) if \( xy < 1 \).

We can express each term in the series in a way that allows us to use this formula. Notice that:

Rewriting the Terms

We can rewrite the term \( \tan^{-1}\left(\frac{2^{k-1}}{1 + 2^{2k-1}}\right) \) as follows:

• Let \( x = 2^{k-1} \) and \( y = 1 + 2^{2k-1} \).
• Then, \( \tan^{-1}\left(\frac{x}{y}\right) = \tan^{-1}\left(\frac{2^{k-1}}{1 + 2^{2k-1}}\right) \).

Identifying a Pattern

By examining the terms, we can see that:

• \( \tan^{-1}\left(\frac{2^{k-1}}{1 + 2^{2k-1}}\right) \) can be expressed in terms of the previous term.
• This suggests a telescoping nature to the series, where many terms will cancel out when summed.

Summing the Series

To find the total sum, we can observe that:

• The first term is \( \tan^{-1}\left(\frac{1}{3}\right) \).
• Each subsequent term can be expressed as a difference of two arctangents, leading to a cancellation effect.

After performing the necessary algebra, we find that the sum converges to:

Final Result

The sum of the series can be expressed as:

\( S_n = \tan^{-1}(1) - \tan^{-1}\left(\frac{1}{2^{n-1}}\right) \)

Thus, as \( n \) approaches infinity, the series converges to \( \frac{\pi}{4} \).

Example Calculation

For instance, if \( n = 3 \):

• We compute \( S_3 = \tan^{-1}(1) - \tan^{-1}\left(\frac{1}{4}\right) \).
• This simplifies to \( \frac{\pi}{4} - \tan^{-1}\left(\frac{1}{4}\right) \).

This approach not only gives us the sum but also illustrates the beauty of the arctangent function and its properties. By recognizing patterns and using identities, we can simplify complex series effectively.