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Problem 1 Easy Difficulty
(a) What is the difference between a sequence and a series?
(b) What is a convergent series? What is a divergent series?
Answer
(a) A sequence is an ordered list of numbers whereas a series is the sum of a list of numbers.
(b) A series is convergent if the sequence of partial sums is a convergent sequence. A series is divergent if it is not convergent.
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Video Transcript
for Part A. We'd liketo list the difference between a sequence and a Siri's. Well, the sequence usually denoted something like this, braces a N. Then you have your starting point, your ending point, or you could denoted by listing its values in order. It's just the order list of numbers, however. A Siri's Siri's will arise from a sequence. So from the sequence, we can go to a new sequence which will call us in and goes from one to infinity. And let's define the ends firm of this new sequence. SN to just be the sum of the first and values of the original sequence. So, for example, as one is a one as to is a one plus a two and so on that here, the Siri's will be the limit as n goes to infinity of s end. So that's the difference is a sequence is this is a list of numbers in order. A Siri's is a limit of a sequence. So for part B was a convergent Siri's well, a Siri's will converge. So let's write it this way. Brackets or braces. Excuse me. SN from one to infinity converges if till limit as and goes to infinity. SN is a real number. So again by this sn, we still mean what we met in part made, that s and is the sum of the first and values. So in other words, this is saying so. This is equivalent to saying the limit and goes to infinity of a one plus dot, dot, dot plus a M. Israel is a real number, and as you could see, is you and go to infinity. This finite sum here in the apprentices well, get closer and closer to the infinite sum. And that's the limit of us. And so this is what it means to converge means a CZ. You take more, more sums. The women exist. Otherwise, the limit of SN is not a real number. So let's just right. It does not exist. Or you could just say its equal two equals infinity or minus infinity and any of these cases. So the limit is not a real number. Then we say diversions, and that's your final answer.
Top Calculus 2 / BC Educators
Grace H.
Numerade Educator
Catherine R.
Missouri State University
Kristen K.
University of Michigan - Ann Arbor
Michael J.
Idaho State University