Help with Fourier Series

[PA3040D]

Dear Expert,
Please check the attached Fourier series solution. Kindly advise if there are any mistakes or steps I have done incorrectly. I would also appreciate your suggestions on how to minimize the time taken to solve it. and what steps that I have done over
Please accept my apologies if the handwriting in the image is not clear enough.

 

[logistic_guy]

Dear Expert,
Please check the attached Fourier series solution. Kindly advise if there are any mistakes or steps I have done incorrectly. I would also appreciate your suggestions on how to minimize the time taken to solve it. and what steps that I have done over
Please accept my apologies if the handwriting in the image is not clear enough.

The graph is not periodic but you wrote it has a period of $\displaystyle 4$. Are you sure that this is the correct graph?

 

[PA3040D]

Of course, yes—that was my mistake during the drawing. However, the solution has been corrected by referring to the original question.
Please take a look at the original question that I have attached.

 

[logistic_guy]

Most of your work is correct. You just need to organize your calculations.

You write $\displaystyle a_n$ when you actually are calculating $\displaystyle a_0$. You write $\displaystyle b_n$ when you are calculating $\displaystyle a_n$ and in the same time you write $\displaystyle b_n$ when you are calculating $\displaystyle b_n$.

You calculated $\displaystyle a_n$ as $\displaystyle +\frac{4}{n^2\pi^2}$ when it should be $\displaystyle -\frac{4}{n^2\pi^2}$ for $\displaystyle n=1,3,5,\cdots$

You say $\displaystyle a_0 = 1$ which is correct. You write:

$\displaystyle f(x) = \frac{a_0}{2} + \cdots \rightarrow$ correct, then you write:

$\displaystyle f(x) = 1 + \cdots \rightarrow$ wrong.

You should write: $\displaystyle f(x) = \frac{1}{2} + \cdots$

Now you know the values of $\displaystyle a_0, a_n,$ and $\displaystyle b_n$. Write the general form as:

$\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}a_n\cos\frac{n\pi x}{2} + b_n\sin \frac{n\pi x}{2}$

Since $\displaystyle a_n$ takes only odd indices, you can change any $\displaystyle n$ there to $\displaystyle 2n-1$. Now replace the coefficients with their values.

$\displaystyle f(x) = \frac{1}{2} + \sum_{n=1}^{\infty}-\frac{4}{(2n-1)^2\pi^2}\cos\frac{(2n-1)\pi x}{2} + (-1)^n\frac{2}{n\pi}\sin \frac{n\pi x}{2}$

Now the second term will keep track of the odd indices while the third term will keep track of both odd and even as $\displaystyle b_n$ takes both.

 

[PA3040D]

Most of your work is correct. You just need to organize your calculations.

You write $\displaystyle a_n$ when you actually are calculating $\displaystyle a_0$. You write $\displaystyle b_n$ when you are calculating $\displaystyle a_n$ and in the same time you write $\displaystyle b_n$ when you are calculating $\displaystyle b_n$.

You calculated $\displaystyle a_n$ as $\displaystyle +\frac{4}{n^2\pi^2}$ when it should be $\displaystyle -\frac{4}{n^2\pi^2}$ for $\displaystyle n=1,3,5,\cdots$

You say $\displaystyle a_0 = 1$ which is correct. You write:

$\displaystyle f(x) = \frac{a_0}{2} + \cdots \rightarrow$ correct, then you write:

$\displaystyle f(x) = 1 + \cdots \rightarrow$ wrong.

You should write: $\displaystyle f(x) = \frac{1}{2} + \cdots$

Now you know the values of $\displaystyle a_0, a_n,$ and $\displaystyle b_n$. Write the general form as:

$\displaystyle f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}a_n\cos\frac{n\pi x}{2} + b_n\sin \frac{n\pi x}{2}$

Since $\displaystyle a_n$ takes only odd indices, you can change any $\displaystyle n$ there to $\displaystyle 2n-1$. Now replace the coefficients with their values.

$\displaystyle f(x) = \frac{1}{2} + \sum_{n=1}^{\infty}-\frac{4}{(2n-1)^2\pi^2}\cos\frac{(2n-1)\pi x}{2} + (-1)^n\frac{2}{n\pi}\sin \frac{n\pi x}{2}$

Now the second term will keep track of the odd indices while the third term will keep track of both odd and even as $\displaystyle b_n$ takes both.

Great thanks sir for help and advice