Could 1-1+1-1+1-1+1-1+... actually converge?

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A series convergence in the normal sense when the limit of partial sums converge. We will commonly say the geometric series converges and the harmonic series diverges because their partial sums do. But Cesàro Summation instead considers the average of the partial sums and asks whether the limit of those converge. For example, 1-1+1-1+1-1... diverges in the normal sense, but because the average of its partial sums is 1/2, it converges in the Cesàro sense. A great example of this is with Fourier Series which experience big jumps near the discontinuities called Gibb's phenomenon. Taking Cesàro sums of the Fourier approximations (called Fejer Sums) instead smooths out Gibb's phenomenon.

0:00 Ordinary Convergence (Review)
3:04 Cesàro Summation
8:18 Fourier Series (Review)
9:28 Cesàro Summation improves Fourier Series
11:28 Brilliant.org/TreforBazett

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