What value does the Fourier series approach at the jumps of the square wave?
At the jump discontinuities, the Fourier series approaches the midpoint of the one-sided limits. For the specified square wave, the left limit is and the right limit is , so the series converges to at the jumps.
Conditions
- The point is a jump discontinuity.
- The one-sided limits exist and are finite.
Reasoning, step by step
- Identify the left-hand limit at the jump.
- Identify the right-hand limit at the jump.
- Calculate the average of the two limits.
- Conclude that the series converges to this average.
Example
The script states: 'At a jump they approach the midpoint, zero in this example.'
Common misconceptions
- Believing the series converges to the function's defined value at the jump.
- Thinking the series diverges at the discontinuity.
Watch the explanation
BilibiliFourier series
0:19 – 0:30Watch this moment ↗
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