Some difficult problems

A DOUBLE INTEGRAL OVIDIU FURDUI Let k ≥ be a natural number and let , i = 1, · · · , k be natural numbers Evaluate ∞ ∞ (e−a1 x − e−a1 y ) · · · (e−ak x − e−ak y ) dxdy (x − y)2 Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008 E-mail address: o0furdui@wmich.edu, ofurdui@yahoo.com A SEQUENCE OF RATIONAL NUMBERS OVIDIU FURDUI Find n→∞ n lim + n + ··· + n n n+1 Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008 E-mail address: o0furdui@wmich.edu, ofurdui@yahoo.com A SIMPLE LIMIT OVIDIU FURDUI Find the limit dx + cos2 x cos2 (2x) · · · cos2 (nx) lim n→∞ Western Michigan University, Math Department, Kalamazoo, MI, 49009 E-mail address: o0furdui@wmich.edu, ofurdui@yahoo.com Date: October 17, 2007 THE LIMIT OF AN INTEGRAL SEQUENCE OVIDIU FURDUI Let a and b be real numbers and let c be a positive number Find the limit b dx c + sin x · sin (x + 1) · · · sin2 (x + n) lim n→∞ a Western Michigan University, Math Department, Kalamazoo, MI, 49009 E-mail address: o0furdui@wmich.edu, ofurdui@yahoo.com Date: October 17, 2007 A QUICKIE OVIDIU FURDUI Prove that ∞ n n=1 n k=1 xk − ln k 1−x =− ln2 (1 − x) , The University of Toledo, Math Department, Toledo, OH E-mail address: ofurdui@yahoo.com, Ovidiu.Furdui@utoledo.edu Date: October 23, 2007 −1 ≤ x < PROPOSED PROBLEM TO CRUX MATHEMATICORUM WITH MATHEMATICAL MAYHEM OVIDIU FURDUI Find the sum ∞ (−1)n−1 n n=1 1− 1 (−1)n−1 + − ··· + n The University of Toledo, Math Department, Toledo, OH E-mail address: ofurdui@yahoo.com, Ovidiu.Furdui@utoledo@edu Date: March 1, 2008 PROPOSED PROBLEM TO THE HARVARD COLLEGE MATHEMATICS REVIEW A TRICKY LIMIT OVIDIU FURDUI Let a, b ≥ be two nonnegative numbers Find the limit n lim n→∞ n+k+b+ k=1 √ n2 + kn + a The University of Toledo, Math Department, Toledo, OH E-mail address: ofurdui@yahoo.com, Ovidiu.Furdui@utoledo@edu Date: September 17, 2007 PROPOSED PROBLEM TO SCHOOL SCIENCE AND MATHEMATICS OVIDIU FURDUI Find the sum ∞ (−1)k ln − k=2 k2 The University of Toledo, Math Department, Toledo, OH E-mail address: ofurdui@yahoo.com, Ovidiu.Furdui@utoledo.edu Date: November 6, 2007 THE LIMIT OF A PRODUCT OVIDIU FURDUI Let α be a real number and let p > Find: n lim n→∞ k=1 np + (α − 1)k p−1 np − k p−1 Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008 E-mail address: o0furdui@wmich.edu, ofurdui@yahoo.com PROPOSED PROBLEM TO REVISTA ESCOLAR DE LA OLIMPIADA IBEROAMERICANA DE MATEMATICA OVIDIU FURDUI Let n be a positive integer Evaluate cos (π {nx}) dx, n where {a} = a − a is the fractional part of a Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008 E-mail address: o0furdui@wmich.edu, ofurdui@yahoo.com Date: November 22, 2006 PROPOSED PROBLEM TO THE COLLEGE MATHEMATICS JOURNAL AN IMPROPER EXPONENTIAL INTEGRAL OVIDIU FURDUI Let n be a fixed nonnegative integer Evaluate ∞ xn ex + + x + x2! + · · · + xn n! dx Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008 E-mail address: o0furdui@wmich.edu, ofurdui@yahoo.com PROPOSED PROBLEM TO CRUX MATHEMATICORUM WITH MATHEMATICAL MAYHEM A QUARTIC FRACTIONAL INTEGRAL OVIDIU FURDUI Evaluate 1 x dx, where {x} = x − x is the fractional part of x Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008 E-mail address: o0furdui@wmich.edu, ofurdui@yahoo.com PROPOSED PROBLEM TO THE MATHEMATICAL GAZETTE OVIDIU FURDUI Let n ≥ and m ≥ be natural numbers Evaluate ∞ ∞ I= (e−nx − e−ny ) (e−mx − e−my ) dxdy (x − y)2 Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008 E-mail address: o0furdui@wmich.edu, ofurdui@yahoo.com A GEOMETRY PROBLEM OVIDIU FURDUI Let ABCD be a convex quadrilateral and let P be a point in the interior of ABCD such that PA = √ AB , PB = √ BC , PC = √ CD 2 and P D = a square E-mail address: ofurdui@yahoo.com √ DA Prove or disprove that ABCD is PROBLEM 11234, THE AMERICAN MATHEMATICAL MONTHLY, 6/113 11234 Let a1 , a2 , · · · , an and b1 , · · · , bn−1 be real numbers with a1 < b1 < a2 < · · · < an−1 < bn−1 < an , and let h be an integrable function from R to R Show that ∞ h −∞ (x − a1 ) · · · (x − an ) (x − b1 ) · · · (x − bn−1 ) ∞ dx = h(x)dx −∞