# hw16solutions - of the problem 2 If the partial sums s n of...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: March 17, 2010 Assignment 16 1. Suppose that none of the numbers a,b,c is a negative integer or zero. Prove that the hypergeo- metric series ab 1! c + a ( a + 1) b ( b + 1) 2! c ( c + 1) + a ( a + 1)( a + 2) b ( b + 1)( b + 2) 3! c ( c + 1)( c + 2) + ... is absolutely convergent for c > a + b and divergent for c < a + b . Here we note that x n +1 x n = ( a + n )( b + n ) ( n + 1)( c + n ) = 1 + ( a + b - c - 1) n + ab - c ( n + 1)( c + n ) . As n → ∞ , the last term tends to 0. Thus, for N suﬃciently large, x n +1 /x n is nonneg- ative for n N . Then, 1 - ± ± ± ± x n +1 x n ± ± ± ± = - ( a + b - c - 1) n + ab - c ( n + 1)( c + n ) , n N. Moroever, n ² 1 - ± ± ± ± x n +1 x n ± ± ± ± ³ c + 1 - a - b. By Corollary 9.2.9, we thus have absolute convergence if c +1 - a - b > 1 and divergence if c +1 - a - b < 1. These latter conditions are equivalent to those iven in the statement
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Unformatted text preview: of the problem. 2. If the partial sums s n of ∑ ∞ n =1 a n are bounded, show that the series ∑ ∞ n =1 a n /n converges to ∑ ∞ n =1 s n /n ( n + 1). Here we apply Abel’s Lemma. We then see that m X n =1 a n n = 1 m s m + m-1 X n =1 ² 1 n-1 n + 1 ³ s n = s m m + m-1 X n =1 1 n ( n + 1) s n . As m → ∞ , we have, by the Pinching principle, s m m → 0 since s m is bounded and 1 /m → 0. What remains after taking this limit is the desired equality. 1...
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