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Let s use the letter i as the counting tag We start at i = 0 and go up to i = n, with each term having the value 1/2i Therefore, the summation notation is
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1/2i
i =0
Limit of a Series
If a series has a limit, we can sometimes figure it out by creating a sequence from the partial sums, and then finding the limit of that sequence
An example Think of the summation in the previous challenge, and imagine what happens as n increases endlessly that is, as n approaches infinity As n grows larger, the sequence of partial sums approaches 2 We can plug the summation into a limit template, and then state that
1/2i = 2
i =0
Another example Let s look once again at the infinite sequence V we saw a little while ago, where the numerators keep alternating between 1 and 1, as follows:
V = 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2 Let s replace every comma by a plus sign, creating the infinite series V+ = 1/2 + 1/2 1/2 + 1/2 1/2 + 1/2 1/2 + We can write this series in summation form as
( 1)i/2
i =1
Now consider the limit of the sequence of partial sums of V+ as the number of terms becomes arbitrarily large We write this quantity symbolically as Lim
( 1) /2
i =1
This limit does not exist, because the sequence of partial sums alternates endlessly between two values, 1/2 and 0
Limit of a Series
Are you confused
Does the combination of limit and summation notation look intimidating Besides getting used to the symbology, you have to keep track of two different indexes, i for the sum and n for the limit It helps if you remember that the two indexes are independent of each other You re finding the limit of a sum as you keep making that sum longer
Here s a challenge!
Find the limit of the partial sums of the infinite series 1/100 + 1/1002 + 1/1003 + 1/1004 + 1/1005 + as the number of terms in the partial sum increases without end That is, find Lim
1/100i
i =1
Solution
In decimal form, 1/100 = 001, 1/1002 = 00001, 1/1003 = 0000001, and so on Let s arrange these numbers in a column with each term underneath its predecessor, and all the decimal points along a vertical line, as the following: 001 00001 0000001 000000001 00000000001
When we look at the series this way, we can see that it must ultimately add up to the nonterminating, repeating decimal 00101010101 From our algebra or number theory courses, we recall that this endless decimal number is equal to 1/99 That s the limit of the sequence of partial sums in the series:
1/100i
i =1
as the positive integer n increases without end It s also the value of the entire infinite series:
1/100i
Sequences, Series, and Limits
Limits of Functions
So far, we ve looked at situations where we move from term to term in a sequence or series Sometimes, such sequences and series have limits (they converge); in other cases they don t have limits (they diverge) Similar phenomena can occur when we have a variable that changes in a smooth, continuous manner, rather than jumping among discrete values
Some functions have limits, and some don t Certain functions increase or decrease without bound, while others reach specific values and stay there Still others increase or decrease continuously without ever passing, or even reaching, a certain value It s also possible for a function to blow up and have no limit at all The solid curve in Fig 19-4 shows the reciprocal function in the first quadrant of the Cartesian plane, where the value of the independent variable is positive The dashed curve shows the negative reciprocal function in the fourth quadrant, where, again, the value of the independent variable is positive The functions are
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