What is the maximum value of sn determine limnrarrinfin sn

Question

Putnam TNG - Limits & Estimation

1: Evaluate lim_x→∞((1/x)(a^x - 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 x_n. Show that x_n < 2/n.

3: Let 0 < a < b. Evaluate

lim_t→0[₀∫¹[bx + a(1 - x)]^t dx]^1/t.

4: Evaluate

lim_n→∞ 1/n⁴_i=1∏²ⁿ(n² + i²)^1/n.

5: Consider the sum

What is the maximum value of S_n? Determine lim_n→∞ S_n.

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 F_n + F_n+1 = F_n+2 for n = 1, 2, 3 . . . . Show that

_n=1∑^∞ 1/F_n < 4.