Let's say we want to approximate a simple square wave (a signal that quickly switches between two values, like a light turning on and off). We can use a trigonometric series to build it.

Step 1: Understand a square wave. It's 1 for half the time and 0 for the other half, then repeats.
---Step 2: The simplest sine wave is sin(x). It looks like a smooth up-and-down curve.
---Step 3: If we add more sine waves with different 'speeds' (frequencies) and 'heights' (amplitudes), we can start to make a shape that looks more like a square wave.
---Step 4: For example, a basic approximation could be (4/pi) * (sin(x) + (1/3)sin(3x) + (1/5)sin(5x) + ...).
---Step 5: Each term (like (1/3)sin(3x)) is a simple sine wave. When you add many of these, the bumps and dips start to flatten out, and the sides become steeper.
---Step 6: The more terms you add, the closer the sum gets to looking exactly like the square wave.
---Answer: By adding an infinite number of these simple sine waves, you can perfectly recreate the sharp corners of a square wave.