Theorem 1 and the Test for the Divergence of Infinite Series

In this video, I go over Theorem 1, which states that the terms of a convergent series approach zero. Conversely, if the terms don't approach zero, the series diverges, hence providing a useful test for divergence. I prove Theorem 1 by re-arranging the n-th term as a difference of two partial sums, which, since they are convergent, are equal and cancel out; thus proving the theorem. I illustrate this with an example. Note that the reverse is not necessarily the case: if the terms approach zero, then the series may converge or may not converge (as in the case of the harmonic series).

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Timestamps

Theorem 1: If a series is convergent, then the limit of the terms approaches zero – 0:00
- Proof: Writing the n-th term as a difference of partial sums – 0:29
- Limit of partial sums converges and cancels out – 3:00
Note 1: Two sequences associated with any series: partial sums and its terms – 4:00
Note 2: If the limit of terms is zero, the series need not be convergent, as in the case of the harmonic series – 4:47
Test for Divergence: If the limit of terms is not zero, then the series diverges – 5:34
Example 8: Show the following series is divergent – 6:58
- Solution: The limit of the terms approaches 1/5, hence the series diverges – 7:25
Note 3: If the limit of the terms is zero, we know nothing about the convergence or divergence of the series – 9:20