Reference no: EM131084687
Putnam TNG - Limits & Estimation
1: Evaluate limx→∞((1/x)(ax - 1/a - 1))1/x, where a > 1.
2: Let n be a positive integer. Starting with the sequence 1, 1/2, 1/3, ..., 1/n, form a new sequence of n - 1 entries 3/4, 5/12, ...,(2n - 1)/2n(n - 1) by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of n - 2 entries, and continue until the final sequence produced consists of a single number xn. Show that xn < 2/n.
3: Let 0 < a < b. Evaluate
limt→0[0∫1[bx + a(1 - x)]t dx]1/t.
4: Evaluate
limn→∞ 1/n4i=1∏2n(n2 + i2)1/n.
5: Consider the sum
What is the maximum value of Sn? Determine limn→∞ Sn.
6: If n is a positive integer prove that
⌊√n + √(n + 1⌋ = ⌊√(4n + 2)⌋
where ⌊x⌋ denotes the greatest integer not exceeding x.
7: The Fibonacci numbers are defined by F1 = 1, F2 = 1 and Fn + Fn+1 = Fn+2 for n = 1, 2, 3 . . . . Show that
n=1∑∞ 1/Fn < 4.
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Calculate the new critical speed
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Analyze the implications of the fall of humanity
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Flip a fair coin until heads occurs twice
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What is the maximum value of sn
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