Definition and Notation of Infinite Series

In this video, I go over infinite series (or just series), which is defined as the summation of the terms of an infinite sequence of numbers. If we add an infinite sequence of numbers whose terms are each greater than 1, then the sum goes to infinity. But if the terms are fractions, then in some cases the sum tends towards a finite limit. In this case, we obtain a convergent series — otherwise, if there is no finite sum, then the series is called divergent. A partial sum is the sum of a specific number of terms of a series. The actual sum is the limit of the partial sums as the terms approach infinity. This is similar to how the improper integral is defined.

#math #calculus #series #sequences #education

Timestamps

• Adding terms of an infinite sequence is called an infinite series or just a series – 0:00
• Adding infinite fractions of factors of 2 – 1:52
• Excel table calculating partial sums of the series and showing it approaches 1: https://1drv.ms/x/s!As32ynv0LoaIh8gwtUIvFFz4HKE0DA – 3:33
• If the partial sum of a series has a limit, then the series has a finite sum – 6:09
• Definition 1: A series is convergent if it has a sum; otherwise, it is divergent – 6:53
• Comparing infinite series to improper integrals – 9:47

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