Cosx+2^2 Cos2x+3^2 Cos3x+-----+n^2CosnxFind sum.... of this series.. by any approach

To find the sum of the series given by the expression \( \cos x + 2^2 \cos 2x + 3^2 \cos 3x + \ldots + n^2 \cos nx \), we can utilize a combination of trigonometric identities and properties of series. This series can be expressed in a more manageable form, and we can derive a formula for its sum.

Breaking Down the Series

The series can be rewritten as:

• First term: \( \cos x \)
• Second term: \( 2^2 \cos 2x \)
• Third term: \( 3^2 \cos 3x \)
• And so on, up to \( n^2 \cos nx \)

Thus, the series can be expressed as:

\( S_n = \sum_{k=1}^{n} k^2 \cos kx \)

Using the Formula for the Sum of Series

To find the sum \( S_n \), we can use a known result from the theory of Fourier series or the properties of generating functions. The sum of the series \( \sum_{k=1}^{n} k^2 \cos kx \) can be derived using the formula:

\( \sum_{k=1}^{n} k^2 e^{ikx} = e^{ix} \frac{d^2}{dx^2} \left( \frac{1 - e^{inx}}{1 - e^{ix}} \right) \)

We can separate the real part to focus on the cosine terms. This involves some calculus, specifically taking derivatives and simplifying the resulting expression.

Calculating the Derivative

Let’s denote:

\( f(x) = \frac{1 - e^{inx}}{1 - e^{ix}}

Taking the first derivative:

\( f'(x) = \frac{d}{dx} \left( \frac{1 - e^{inx}}{1 - e^{ix}} \right)

Using the quotient rule, we can find \( f'(x) \) and then take the second derivative to find \( f''(x) \). This process will yield a more complex expression, but ultimately, we can extract the cosine terms from the exponential form.

Final Expression for the Sum

After performing the necessary calculations, the sum can be expressed in a closed form. The final result for the sum \( S_n \) can be derived as:

\( S_n = \frac{n \sin\left(\frac{nx}{2}\right) \cos\left(\frac{(n+1)x}{2}\right)}{\sin\left(\frac{x}{2}\right)^3}

This formula provides a compact way to compute the sum of the series for any integer \( n \) and angle \( x \).

Example Calculation

Let’s say we want to calculate the sum for \( n = 3 \) and \( x = \frac{\pi}{4} \):

\( S_3 = \frac{3 \sin\left(\frac{3\pi}{8}\right) \cos\left(\frac{4\pi}{8}\right)}{\sin\left(\frac{\pi}{8}\right)^3}

By substituting the values and simplifying, we can find the numerical value of the sum.

Conclusion

This approach not only provides the sum of the series but also illustrates the power of calculus and trigonometric identities in solving complex series. Understanding these concepts can greatly enhance your mathematical toolkit.