Theorem 2: Series of Sums is equal to Sum of Series

https://play.3speak.tv/embed?v=mes/3vrykh9o

In this video, I go over Theorem 2, which states that we can move a constant out of the sum of a series as well as add or subtract series individually. These follow from the Limit Laws of Sequences, and I use this to prove that the series involving an addition of terms is equal to an addition of the individual series. I go over an example, as well as make a note that a finite number of terms does not affect the convergence or divergence of a series.

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#math #calculus #series #sequences #education

Timestamps

• Theorem 2: Adding and subtracting convergent series – 0:00
• Proof of part ii of Theorem 2 via Limit Laws for Sequences – 1:54
• Taking the limit as the sum of partial sums approaches infinity – 3:55
• The series of sums is equal to the sum of the series – 6:23
• Example 9: Series of sums – 7:39
• Solution: The second series is a convergent geometric series that sums to 1 – 8:03
  • The first series was solved from Example 6 via the telescoping sum and is equal to 3 – 9:42
  • Adding the series obtains 3 + 1 = 4 as the sum – 11:00
• Note 4: A finite number of terms doesn't affect the convergence or divergence of a series – 11:40

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