Unformatted text preview: = x n +2 = Â·Â·Â· = x m = y := 1 n n âˆ‘ k =1 x k . #4. (a). Show that the property (1) holds true for f ( x ) := e x . (b). Use the inequality (2) with f ( x ) := e x for the proof of the inequality ( a 1 a 2 Â·Â·Â· a n ) 1 /n â‰¤ 1 n ( a 1 + a 2 + Â·Â·Â· + a n ) for all natural n and a k â‰¥ . #5. Using the fact (Problem #4 in Homework #9) that s n := 1 + 1 2 + 1 3 + Â·Â·Â· + 1 nln n â†’ Î³ = const as n â†’ âˆž , evaluate the sum of the series âˆž âˆ‘ n =1 (1) n1 n = 11 2 + 1 3 Â·Â·Â· . Hint. Express Ïƒ 2 n := 11 2 + 1 3 Â·Â·Â· + 1 2 n11 2 n in terms of s 2 n and s n ....
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 Fall '09
 Calculus, Real Numbers, Limit, lim

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