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# Fourier coefficients for sine terms

Fourier coefficients for sine terms. Created by Sal Khan.

## Want to join the conversation?

• Why can we be sure that the integral of an infinite sum is the infinite sum of the integrals? I know it is clear from the finite case, but is it still true for the infinite case?
• According to me, an infinite sum is actually nothing but an infinitely countable sum. To better visualize the scenario, consider sum of the integral of 10 elements (a simple math problem) in comparison to 1 million elements (in the case of images consisting of millions of pixels). The earlier case can be better related to the finite case wherein the elements are finite, but the latter case of 1 million elements is 'RELATIVELY' infinite with respect to the former one. All in all the definition remains true in either case. For generality we need to take the case of infinity, however, in reality everything is countable even though it leads to infinity. I hope I helped solve your problem. If I am wrong, please do acknowledge me regarding the same...!
• i have a bit of a 'chicken and egg' question.

i understand the toolkit sal assembled and how many of the integrals evaluate to 0.

but, what is the motivation for integrating f(t) by itself to find a_0 and then f(t)cos(nt) to find a_n then f(t)sin(nt) to find b_n.

do we do this because of our toolkit and convenience (already knowing that most of our terms will = zero)? or is there another reason?
• I think what you are asking is... Is Sal presenting a linear line of reasoning toward the goal, or, does he (knowing the outcome) first develop a useful toolkit, followed by application of the tools?

The answer is the latter. The first set of videos develops a toolkit that exploits lots of trig properties and integration and cancellation to generate many 0's. What's left behind are some relatively simple and very general expressions for the a_n and b_n terms for any f(t).

In the final video (the next one), he puts it all together and does the Fourier series for a square wave. With all the 'tools' available he can do this in just a few minutes rather than slogging through the previous half hour of videos for just this one example.

All those derivations are actually very important to your understanding of signal processing. It is a very valuable exercise to look at those calculus expressions and draw each one in the time domain and the frequency domain. These derivations describe how AM modulation works, the theory behind the superheterodyne receiver, and pretty much everything else in signal processing (analog and digital).