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To calculate the value of using equation (21), we need to use a numerical method to estimate the sum of the series . However, this series is infinite and we need to truncate it somehow. One way to do this is to approximate the parameter with a rational number , where and are integers. Then, we can use equation (17) to rewrite the series as a finite sum: (34)

The advantage of using this method is that we can control the accuracy of the approximation by choosing the values of and . The smaller the denominator , the closer the rational number is to the parameter . However, this also means that the finite series will have more terms to sum up, which increases the computational cost. Therefore, we need to balance the trade-off between accuracy and efficiency.

To find a suitable rational approximation for , we can use a technique called continued fraction expansion. This technique expresses a real number as a nested fraction of integers, such as: (35) where , , , ... are positive integers. The continued fraction can be truncated at any stage to obtain a rational approximation for . For example, if we truncate after the first term, we get: (36) which is a good approximation for .

We can use an algorithm called Euclid's algorithm to find the continued fraction expansion of any real number. The algorithm works by repeatedly applying the division algorithm to the numerator and denominator of the fraction. For example, to find the continued fraction expansion of , we start with: (37) Then, we apply the division algorithm to get: (38) where is the quotient and is the remainder. We repeat this process with and until we get a zero remainder. The quotients are then the coefficients of the continued fraction expansion.

Let us see an example of Euclid's algorithm applied to . We start with: (39) Then, we apply the division algorithm to get: (40) where and . We repeat this process with and to get: (41) where and . We continue with and to get: (42) where and . Finally, we get a zero remainder with and : (43) where and . Therefore, the continued fraction expansion of is: (44)

Now that we have the continued fraction expansion of , we can truncate it at any stage to obtain a rational approximation for . The more terms we include, the better the approximation. For example, if we truncate after the first term, we get: (45) which is the same as equation (36). If we truncate after the second term, we get: (46) which is a better approximation than equation (45). If we truncate after the third term, we get: (47) which is even better than equation (46). And so on.

We can use a formula to find the rational approximation for any truncation of the continued fraction expansion. The formula is: (48) where and are the numerator and denominator of the rational approximation, and are the coefficients of the continued fraction expansion, and is the index of truncation. For example, if we truncate after the second term, we get: (49) which is the same as equation (46). If we truncate after the third term, we get: (50) which is the same as equation (47). And so on. 061ffe29dd