Ser_hnd - Convergent or Divergent Series Definition The...

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Convergent or Divergent Series Definition: The harmonic series is the divergent series 1 1 1 2 1 3 1 1 n n n = + + + ⋅⋅⋅ + + ⋅⋅⋅ = Theorems: I. Telescoping Series If a n is a telescoping series with a b b n n n = - + 1 , then a n converges iff { } b n converges. Furthermore, if { } b n converges to L , then a n converges to b L 1 - . II. Geometric Series Let a 0 . The geometric series ar a ar ar ar n n n - - = = + + + ⋅⋅⋅ + + ⋅⋅⋅ 1 2 1 1 (i) converges and has the sum S a r = - 1 if r < 1 (ii) diverges if r 1 III. If a series a n is convergent, then lim n n a →∞ = 0 . IV. nth-Term Series (i) If lim n n a →∞ 0 , then the series a n is divergent. (ii) If lim n n a →∞ = 0 , then further investigation is necessary to determine whether the series a n is convergent or divergent. V. If a n and b n are series such that a b j j = for every
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Unformatted text preview: j k where k is a positive integer, then both series converge or both series diverge. VI. For any positive integer k , the series a a a n n = + + ⋅⋅⋅ = ∞ ∑ 1 2 1 and a a a n k k n = + + ⋅⋅⋅ + + = ∞ ∑ 1 2 1 either both converge or both diverge. VII. If a n ∑ and b n ∑ are convergent series with sums A and B , respectively, then (i) ( ) a b n n + ∑ converges and has sum A B + (ii) ca n ∑ converges and has sum cA for every real number c (iii) ( ) a b n n-∑ converges and has sum A B-VIII. If a n ∑ is a convergent series and b n ∑ is divergent, then the series ( ) a b n n + ∑ is divergent....
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