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# series - INFINITE SERIES(Chapter 11 General facts A...

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INFINITE SERIES (Chapter 11) General facts: A sequence is nothing but an infinite list. A series is a sum of infinitely many numbers. An infinite series may either converge (i.e. reach a finite total) or diverge (i.e. not reach a finite total). A famous example of a convergent series is any geometric series with | r | < 1 (e.g. r = 1 2 ): X n =1 1 2 n = 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + · · · converges A famous example of a divergent series is the harmonic series: X n =1 1 n = 1 1 + 1 2 + 1 3 + 1 4 + 1 5 + · · · diverges These are two very famous series, so remember them. (To prove that these two series converge and diverge respectively, you have to do a bit of algebra.) So, if a n = 1 2 n , the series X a n converges. If a n = 1 n , the series X a n diverges. VERY IMPORTANT: The above shows that if a n approaches 0, this does NOT GUARANTEE that the series X a n converges! If a n does NOT approach 0, then the series X a n MUST DIVERGE. If a n DOES approach 0, then the series X a n MIGHT CONVERGE or MIGHT DIVERGE. More work is needed to decide which! You’re not done! Rough idea: If a n approaches 0 “fast enough”, the series X a n converges. If a n doesn’t approach 0 “fast enough”, the series X a n diverges.

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In the MOST COMMON TYPE OF EXAM PROBLEM on infinite series, you will be given an infinite series X a n where the formula for a n might be complicated. You will have to decide (with proof) whether the series converges or diverges. Two famous series: GEOMETRIC SERIES. Any infinite series of the form a + ar + ar 2 + ar 3 + ar 4 + · · · = X n =0 ar n = X n =1 ar n - 1 is convergent if | r | < 1, and is divergent if | r | ≥ 1. If it converges, its sum is a 1 - r .
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series - INFINITE SERIES(Chapter 11 General facts A...

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