How to evaluate:

Summation r=1 to n Cosec^-1(4r⁴+1)^1/2

To evaluate the summation from \( r = 1 \) to \( n \) of \( \text{Cosec}^{-1}(4r^4 + 1) \), we need to break down the components involved in the expression. Let's first clarify what \( \text{Cosec}^{-1}(x) \) means. The cosecant inverse function, or arcsin, gives us the angle whose cosecant is \( x \). In simpler terms, if \( y = \text{Cosec}^{-1}(x) \), then \( \csc(y) = x \), which implies \( y = \frac{1}{\sin(y)} \). This means we can express \( \text{Cosec}^{-1}(4r^4 + 1) \) in terms of sine.

Breaking Down the Expression

We can rewrite the cosecant inverse function as:

• Step 1: Recognize that \( \text{Cosec}^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \).
• Step 2: Therefore, \( \text{Cosec}^{-1}(4r^4 + 1) = \sin^{-1}\left(\frac{1}{4r^4 + 1}\right) \).

Setting Up the Summation

Now, we can express the summation as:

\( \sum_{r=1}^{n} \sin^{-1}\left(\frac{1}{4r^4 + 1}\right) \)

This summation involves evaluating the arcsine function for each integer \( r \) from 1 to \( n \). The next step is to analyze how this function behaves as \( r \) increases.

Behavior of the Function

As \( r \) increases, the term \( 4r^4 + 1 \) grows significantly. Consequently, \( \frac{1}{4r^4 + 1} \) approaches zero, which means that \( \sin^{-1}\left(\frac{1}{4r^4 + 1}\right) \) approaches zero as well. This suggests that the terms in the summation will become very small for larger values of \( r \).

Approximation for Large n

For large \( n \), we can approximate the summation. The arcsine function behaves like its argument when the argument is small. Therefore:

\( \sin^{-1}(x) \approx x \) when \( x \) is close to zero.

Applying this approximation, we have:

\( \sin^{-1}\left(\frac{1}{4r^4 + 1}\right) \approx \frac{1}{4r^4 + 1} \)

Final Summation

Thus, the summation can be approximated as:

\( \sum_{r=1}^{n} \frac{1}{4r^4 + 1} \)

This summation can be evaluated using techniques from calculus or numerical methods, especially for large \( n \). However, for practical purposes, you may want to compute the first few terms directly to get an idea of the behavior of the series.

Example Calculation

Let’s compute the first few terms:

• For \( r = 1 \): \( \frac{1}{4(1)^4 + 1} = \frac{1}{5} \)
• For \( r = 2 \): \( \frac{1}{4(2)^4 + 1} = \frac{1}{65} \)
• For \( r = 3 \): \( \frac{1}{4(3)^4 + 1} = \frac{1}{325} \)

Adding these terms gives a rough estimate of the summation for small \( n \). As \( n \) increases, the contributions from each additional term diminish rapidly.

Conclusion

In summary, evaluating the summation \( \sum_{r=1}^{n} \text{Cosec}^{-1}(4r^4 + 1) \) involves understanding the behavior of the cosecant inverse function and approximating it for large \( n \). The series converges slowly, and for practical calculations, focusing on the first few terms can provide useful insights into its value.