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401 Calc II
9.2 Series and Convergence
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Def: An Infinite Series is the sum of the infinite number of terms in a sequence.
∞
∑ 𝑎𝑛 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯ 𝑎𝑛 + ⋯
𝑛=1
We will be interested in discovering the sum of the Infinite series. In some cases the sum will actually converge
to a finite number, in which case we say the series converges. Of course in other cases the series may diverge.
At this point one needs to be clear of the difference between a sequence and a series. A sequence is an ordered
list of numbers 𝑎1 , 𝑎2 , … while an infinite series is a sum of numbers 𝑎1 + 𝑎2 + ⋯.
To determine if a series converges or diverges we will consider the sequence of partial sums.
Def: A partial sum is the sum of a finite number of terms in a sequence.
1
𝑆1 = ∑ 𝑎𝑛 = 𝑎1
𝑛=1
2
𝑆2 = ∑ 𝑎𝑛 = 𝑎1 + 𝑎2
𝑛=1
3
𝑆3 = ∑ 𝑎𝑛 = 𝑎1 + 𝑎2 + 𝑎3
𝑛=1
…
3
𝑆𝑛 = ∑ 𝑎𝑛 = 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛
𝑛=1
401 Calc II
9.2 Series and Convergence
Page 2 of 3
Geometric Series
There are many different ways to categorize series. One such category and one of the main focuses on this topic
in this book is that of Geometric Series.
𝑛
Def: A geometric series is any series given by ∑∞
𝑛=0(𝑎𝑟 ) where 𝑎 is a nonzero constant and 𝑟 is a ratio (fraction)
∞
∑ 𝑎 ∙ 𝑟 𝑛 = 𝑎 + 𝑎𝑟 + 𝑎𝑟 2 + 𝑎𝑟 3 + ⋯ + 𝑎𝑟 𝑛 + ⋯
𝑛=0
Note each term is a constant multiple of the previous term.
For a geometric series we can actually find the sum IF 0 < |𝑟| < 1.
If we have time, I want to go back and prove theorem 9.6.
401 Calc II
9.2 Series and Convergence
Page 3 of 3