Let S(n) = f(1)+ f(2) + ·…+ f(n). For each f(n) below, derive the answer to the question S(n)= 0(?). 3. Simplify your answer as much as you can. 3 (a) f(n) = 1/(2n +3) (c) f(n) = 2" log ³ n (b) f(n) = n² log³n (d) f(n) = 2" n³ log 3 (e) f(n) = 2" / log n (f) f(n) = 2\n log n (g) f(n) = nvñ (i) f(n) = n log n (k) f(n) = (1 + l/n)r² (m) Consider any super-polynomial sub-exponential function f(n) = 2"/8(") where g(n) is a SAF and g(n) e a(1) n o(n). Under what extra conditions, if any, do we have S(n) = © (g(n) f(n)) ? (h) f(n) = (log n)" (1) f(n) = n log log n (1) f(n) = (1 + 1/n)n!.5
Let S(n) = f(1)+ f(2) + ·…+ f(n). For each f(n) below, derive the answer to the question S(n)= 0(?). 3. Simplify your answer as much as you can. 3 (a) f(n) = 1/(2n +3) (c) f(n) = 2" log ³ n (b) f(n) = n² log³n (d) f(n) = 2" n³ log 3 (e) f(n) = 2" / log n (f) f(n) = 2\n log n (g) f(n) = nvñ (i) f(n) = n log n (k) f(n) = (1 + l/n)r² (m) Consider any super-polynomial sub-exponential function f(n) = 2"/8(") where g(n) is a SAF and g(n) e a(1) n o(n). Under what extra conditions, if any, do we have S(n) = © (g(n) f(n)) ? (h) f(n) = (log n)" (1) f(n) = n log log n (1) f(n) = (1 + 1/n)n!.5
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Solve (a) and (f) please
Transcribed Image Text:Let S(n)= f(1)+ f(2) + ·…·+ f(n).
For each f(n) below, derive the answer to the question S(n)= ©(?).
3.
Simplify your answer as much as you can.
(a) f(n) = 1/(2n +3)
(c) f(n) = 2" log ³ n
(b) f(n) = n² log ³ n
(d) f(n) = 2" n³ log n
3
(e) f(n) = 2" / log n
(f) f(n) = 2\n log n
(g) f(n) = nvñ
(i) f(n) = n log n
(k) f(n) = (1 + 1/nyr?
(m) Consider any super-polynomial sub-exponential function f(n) = 2"/8(n)
where g(n) is a SAF and g(n) e a(1) n o(n).
Under what extra conditions, if any, do we have S(n) = © (g(n) f(n)) ?
(h) f(n) = (log n)"
(j) f(n) = n log log n
(1) f(n) = (1 + 1/nyr!.5
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