6 If a seauenu {ak} converges, then there is a number L such that all but a finite number of the...

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6 If a seauenu {ak} converges, then there is a number L such that all but a finite number of the terms of {ak} lie within 1 of L.
n 7 If all partial sums Sn = I ak of any series are less than some constant k=1 L > 0, then the series converges.
co 8 If a series I ak converges, then its partial sums Sn are bounded. That k=1 is, there are constants m and M such that m
9 Beginning at some index value k = n, a convergent sequence {ak} is always
bounded.
10. A bounded sequence (ak} is always convergent.

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David · Accepted Answer

We wish to determine whether each of the following statements is true or false.
6. If the sequence  ka  converges, then there is a number L such that all but a finite number 
of terms of  ka  lie within 1 of L. 
 
This is true. Suppose .ka L  Then for all 0,   in particular for 1,   there exists a 
positive number N such that 1ka L     whenever .k N  Since all but finitely many 
elements of the sequence  ka  have indices ,

6 If a seauenu {ak} converges, then there is a number L such that all but a finite number of the terms of {ak} lie within 1 of L. n 7 If all partial sums Sn = I ak of any series are less than some...

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