12-3 INFINITE SEQUENCES AND SERIES
Jessica P.
Honors Introduction to College Math (ICM)
Vocabulary that you will need to know:
| Infinite Sequence: | This type of sequence can have an infinite amount of terms. As n increases, the terms of the sequence decrease and get closer and closer to zero. |
| Limit: | A point in which a sequence or series can not go above or below. |
| Sum of an Infinite Series: | If Sn is the sum of n terms of a series, and S is a number such that S > Sn for all n, and S - Sn approaches zero as n increases without limit, then the sum of the infinite series is S. |
| Sum of an Infinite | The sum, S, of an infinite geometric series for which the absolute value of r < 1 is given by the following formula. S = a1 / 1 - r |
HOW TO SOLVE A LIMIT PROBLEM:
| Evaluate expression: | lim n --> 6n + 3 / n |
| 1st Step: Simplify Expression: | lim n --> ( 6 + 3/n ) The "n" from the number 6 cancels out. |
| 2nd Step: The limit of a sum equals the sum of the limits. | lim n --> 6 + lim --> 3/n
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| 3rd Step: | The lim n--> 3/n = 0. The only thing left is 6, therefore the limit is 6. |
HOW TO SOLVE AN INFINITE GEOMETRIC SERIES PROBLEM:
| Find the sum of the series: | 2, 1, .5, . . . |
| Find the ratio of the series: | To find the ratio: .5/1 = .5 & 1/2 = .5 The common ratio is .5. Since the ratio is less than 1 we can use the formula and proceed. |
| Plug the parts that you know into the formula and solve: | We know the formula for finding the sum of an infinite geometric series is S = a1 / 1- r. A1 is the first term in the series and in this case is 17.5 and r is the ratio, .5. Therefore we plug in: S = 17.5 / 1-.5 and solve. The next step would be S = 17.5 / .5. So the sum of this infinite geometric series is 35. |
For more information on this topic, refer to: http://www.richland.cc.il.us/james/lecture/m116/sequences/geometric.html
CONGRATULATIONS!! YOU HAVE SUCCESSFULLY COMPLETED THE SECTION ON INFINITE SEQUENCES AND SERIES!!
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