Certainly! Let’s examine the given series and determine the total sum step by step.
The series in question is given as:
[tex]\[
S = \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{x}{13}\right) + \tan^{-1}\left(\frac{1}{21}\right) + \ldots
\][/tex]
To solve this, let's assume the series follows a particular pattern in its terms. Often, in such problems, each term takes the form [tex]\(\tan^{-1}\left(\frac{1}{2n+1}\right)\)[/tex] for successive values of [tex]\(n\)[/tex].
Let’s compute a few initial values to understand the pattern better:
1. First term: [tex]\(\tan^{-1}(1/1) = \tan^{-1}(1)\)[/tex]
2. Second term: [tex]\(\tan^{-1}(1/3)\)[/tex]
3. Third term: [tex]\(\tan^{-1}(1/5)\)[/tex]
We use the identity for the sum of arctangents:
[tex]\[
\tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left(\frac{a + b}{1 - ab}\right)
\][/tex]
Applying this step-by-step:
### Step 1: Sum of the first two terms
[tex]\[
\tan^{-1}(1) + \tan^{-1}(1/3)
\][/tex]
For the specific values:
[tex]\[
a = 1, \quad b = 1/3
\][/tex]
Using the identity:
[tex]\[
\tan^{-1}(1) + \tan^{-1}(1/3) = \tan^{-1}\left(\frac{1 + 1/3}{1 - 1 \cdot 1/3}\right) = \tan^{-1}\left(\frac{1.3333}{0.6667}\right) = 2.0
\][/tex]
Hence:
[tex]\[
\tan^{-1}(1) + \tan^{-1}(1/3) \approx \tan^{-1}(2)
\][/tex]
### Step 2: Adding the third term
[tex]\[
\tan^{-1}(2) + \tan^{-1}(1/5)
\][/tex]
For the specific values:
[tex]\[
a = 2, \quad b = 1/5
\][/tex]
Using the identity:
[tex]\[
\tan^{-1}(2) + \tan^{-1}(1/5) = \tan^{-1}\left(\frac{2 + 1/5}{1 - 2 \cdot 1/5}\right) = \tan^{-1}\left(\frac{2.2}{0.6}\right)
\][/tex]
Hence:
[tex]\[
\tan^{-1}(2) + \tan^{-1}(1/5) \approx \tan^{-1}(3.6667)
\][/tex]
### Infinite Sum Consideration
As the number of terms increases, the series converges. The infinite series [tex]\(\sum_{n=0}^\infty \tan^{-1}\left(\frac{1}{2n+1}\right)\)[/tex] is known to converge to [tex]\(\frac{\pi}{4}\)[/tex].
Thus, the summation [tex]\( S \)[/tex] converges to:
[tex]\[
S = \frac{\pi}{4}
\][/tex]
### Conclusion
Finally, taking the tangent of [tex]\(S\)[/tex]:
[tex]\[
\tan(S) = \tan\left(\frac{\pi}{4}\right)
\][/tex]
We know:
[tex]\[
\tan\left(\frac{\pi}{4}\right) = 1
\][/tex]
Therefore:
[tex]\[
\tan(S) = 1
\][/tex]
