Sequences, Series, and Limits

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the second series, we add up 800 ai s to get the final sum c, and the counting tag i goes from 1 to 800 In the third series, we add up infinitely many ai s to get the final sum d, and the counting tag i ascends through the entire set of positive integers Suppose that we have a series with n terms, as follows: a1 + a2 + a3 + + an 2 + an 1 + an = k In this case, we add up n ai s to get the final sum k

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The big sigma Let s go back to the series with eight terms We can write it down in a cryptic but informationdense manner as

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ai = b

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i =1

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We read this expression out loud as, The summation of the terms ai, from i = 1 to 8, is equal to b The large symbol is the uppercase Greek letter sigma, which stands for summation or sum Now let s look at the series in which 800 terms are added:

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ai = c

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We can read this aloud as, The summation of the terms ai, from i = 1 to 800, is equal to c In the third example containing infinitely many terms, we can write

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ai = d

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i =1

This statement can be read as, The summation of the terms ai, from i = 1 to infinity, is equal to d Finally, in the general case, we can write

ai = k

i =1

and read it aloud as, The summation of the terms ai, from i = 1 to n, is equal to k

A more sophisticated example Suppose we want to determine the value of an infinite series starting with 1, then adding 1/2, then adding 1/4, then adding 1/8, and going on forever, each time cutting the value in half As things work out, we get

1 + 1/2 + 1/4 + 1/8 + = 2 even though the series has infinitely many terms We can also write 1/20 + 1/21 + 1/22 + 1/23 + = 2 In summation notation, we write

1/2i = 2

i =0

Summation Shorthand

Are you confused

If you re baffled by the idea that we can add up infinitely many numbers and get a finite sum, you can use the frog-and-wall analogy Imagine that a frog sits 8 meters (8 m) away from a wall Then she jumps halfway to the wall, so she s 4 m away from it Now imagine that she continues to make repeated jumps toward the wall, each time getting halfway there (Fig 19-3) No finite number of jumps will allow the frog to reach the wall To accomplish that goal, she would have to take infinitely many jumps This scenario can be based on a sequence of partial sums of a series S = 4 + 2 + 1 + 1/2 + 1/4 + 1/8 + A real-world frog cannot reach the wall by jumping halfway to it, over and over But in the imagination, she can There are two ways this can happen First, in the universe of mathematics, we have an infinite amount of time, so an infinite number of jumps can take place Another way around the problem is to keep halving the length of time in between jumps, say from 4 seconds to 2 seconds, then to 1 second, then to 1/2 second, and so on This will make it possible for our cosmic superfrog to hop an infinite number of times in a finite span of time Either way, when she has finished her journey and her nose touches the wall, she ll have traveled exactly 8 m Therefore, the sum total of the lengths of her jumps is S = 4 + 2 + 1 + 1/2 + 1/4 + 1/8 + = 8

Here s a challenge!

Consider the series that we dealt with in A more sophisticated example a couple of paragraphs ago, but only up to the reciprocal of the nth power of 2 Let Sn be the partial sum of this series up to, and including, that term Write Sn in summation notation