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As n grows larger without end, so does n /2 Therefore Lim n /2
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is undefined, so we know that Lim n 2/(2n)
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is also undefined Alternatively, we can say that this limit doesn t exist, or that it s meaningless
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Sequences, Series, and Limits
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By now, you should suspect that any given sequence must fall into one or the other of two categories: convergent (meaning that it has a limit) or divergent (meaning that it doesn t have a limit) But what if a sequence alternates between two numbers endlessly Once again, look at the sequence H = 1, 1, 1, 1, 1, 1, We might be tempted to suggest that a sequence of this type has two different limits, but it doesn t converge on any single number However, that won t work because a limit must always be a single value that we can specify as a number In cases like this, it s customary to say that the limit is not defined
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Here s a challenge!
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Consider the sequence D in which the numerators alternate between 1 and 1, while the denominators start at 1 and increase by 1 with each succeeding term For any positive integer n, the nth term is ( 1)n/n, so that D = 1/1, 1/2, 1/3, 1/4, 1/5, , ( 1)n/n, Does this sequence have a limit If so, what is it If not, why not
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Solution
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As n becomes extremely large, the absolute value of the numerator is always 1, although the sign alternates The denominator increases steadily, and without end If we choose a tiny positive or negative real number, we can always find a term that s closer to 0 than that number, but we ll never actually reach 0 from either the positive side or the negative side Therefore Lim ( 1)n/n = 0
Here s another challenge!
Consider the following sequence: K = ( 1 1/1), (1 + 1/2), ( 1 1/3), (1 + 1/4), , [( 1)n + ( 1)n/n], The parentheses and brackets are not technically necessary here, but they visually isolate the terms from one another Does K have a limit If so, what is it If not, why not
Solution
Each term in K is expressed as a sum The first addend alternates between 1 and 1, endlessly The second addend is identical to the corresponding term in the sequence D that we evaluated in the previous challenge We determined that D converges toward 0 The terms in K therefore approach
Summation Shorthand
two different values, 1 and 1, as we generate terms indefinitely If we want to claim that a sequence has a limit, we must take that expression literally A limit means one and only one limit We therefore conclude that Lim ( 1)n + ( 1)n/n
is not defined
Summation Shorthand
Mathematicians have a shorthand way to denote long sums This technique can save a lot of space and writing time We can even write down an infinite sum in a compact statement It s called summation notation
Specify the series Imagine a set of constants, all denoted by a with a subscript, such as
{a1, a2, a3, a4, a5, a6, a7, a8} Suppose that we add up the elements of this set, and call the sum b We can write this sum out term by term as a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 = b That s easy because we have only eight terms, but if the set contained 800 elements, writing down the entire sum would be exasperating We could put an ellipsis in the middle of the sum, calling it c and then writing a1 + a2 + a3 + + a798 + a799 + a800 = c If the series had infinitely many terms, we could use an ellipsis after the first few terms and leave the statement wide open after that, calling it d and then writing a1 + a2 + a3 + a4 + a5 + = d
Tag the terms Let s invent a nonnegative-integer variable and call it i Written as a subscript, i can serve as a counting tag in a series containing a large number of terms Don t confuse this i with the symbol some texts use to represent the unit imaginary number, which is the positive square root of 1! In the above-described series, we can call each term by the generic name ai In the first series, we add up eight ai s to get the final sum b, and the counting tag i goes from 1 to 8 In
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