Homework Solution: Can someone help with number 8?…

    Can someone help with number 8? 7. For each assertion in 7(a)-(c), prove the assertion directly from the definition of the big-O asymptotic notation if it is true by finding values for the constants c and no. On the other hand, if the assertion is false, give a counter-example. Then answer the question in 7(d) F denotes the set of all functions from Zt to R (a) Let f(n):Z+→ R+. DEFINITION 1. A relation on a set is reflexive if each element is related to itself Assertion: The relation is big-O of is reflexive over F, In other words, f(n) E O(f(n)). [5 points] (b) Let f(n): Z+ → R+ and g(n) : Z+ → R+. DEFINITION 2. A relation on a set is antisymmetric if whenever an element X is related to an element Y and Y is related X, then X = Y. Assertion: The relation is big-O of is antisymmetric over F. In other words, if f(n) E O(g(n)) and g(n) O(f(n)), then f(n) - g(n). [5 points] Let e(n): Z+ → R+, f(n) : Z+ → R+ and g(n) : Z+ → R+. DEFINITION 3. A relation on a set is transitive if whenever an element X is related to Y and Y is related Z, then X is related to Z Assertion: The relation is big-O of is transitive over F. In other words, if e(n) E O(f(n)) and f(n) O(g(n)), then e(n) O(g(n)). [5 points] (c) (d) Is is big-O of a partial order on F? [5 points] DEFINITION 4. A relation is a partial order on a set if it is reflexive, antisymmetric and transitive 8. Given an infinite series s = Σ f(n), where f(n) is a continuous positive monotonically de- creasing function that converges and n EZ+, s can be bounded, using improper integrals, as follows k+1 Using the inequality in (1) and 5, prove that Σ E Θ(1). [10 points]
    For each assertion in 7(a)-(c), prove the assertion directly from the definition of the big-O asymptotic notation if it is true by finding values for the constants c and n_0. On the other hand, if the assertion is false, give a counter-example. Then answer the question in 7(d). F denotes the set of all functions from Z^+ to R^+. (a) Let f(n): Z^+ rightarrow R^+. A relation on a set is reflexive if each element is related to itself. The relation "is big-O of" is reflexive over F, In other words, f(n) elementof O(f(n)). (b) Let f(n): Z^+ rightarrow R^+ and g(n): Z^+ rightarrow R^+. A relation on a set is antisymmetric if whenever an element X is related to an element Y and Y is related X, then X = Y. The relation "is big-O of" is antisymmetric over F. In other words, if f(n) elementof O(g(n)) and g(n) elementof O(f(n)), then f(n) = g(n). (c) Let e(n): Z^+ rightarrow R^+, f(n): Z^+ rightarrow R^+ and g(n): Z^+ rightarrow R^+. A relation on a set is transitive if whenever an element X is related to Y and Y is related Z, then X is related to Z. The relation "is big-O of" is transitive over F. In other words, if e(n) elementof O(f(n)) and f(n) elementof O(g(n)), then e(n) elementof O(g(n)). (d) Is "is big-O of" a partial order on F? A relation is a partial order on a set if it is reflexive, antisymmetric and transitive. Given an infinite series s = sigma^infinity_n = 1 f(n), where f(n) is a continuous positive monotonically decreasing function that converges and n elementof Z^+, s can be bounded, using improper integrals, as follows: sigma^k_n = 1 f(n) + integral^infinity_k + 1 f(n) dn lessthanorequalto s lessthanorequalto sigma^k_n = 1 f(n) + integral^infinity_k f(n) dn, k elementof Z^+ Using the inequality in (1) and k = 5, prove that sigma^infinity_n = 1 1/n^3 elementof theta (1).

    Expert Answer

     
    $$int_{5}^{infty} frac{1}{n^3} dn$$ converges to a con

    Can someone succor with compute 8? 7. Coercion each assumption in 7(a)-(c), assay the assumption instantly from the determination of the big-O asymptotic notation if it is gentleman by opinion values coercion the fixeds c and no. On the other artisan, if the assumption is bogus, afford a counter-example. Then solution the inquiry in 7(d) F denotes the firm of unimpaired exercises from Zt to R (a) Suffer f(n):Z+→ R+. DEFINITION 1. A kindred on a firm is mutual if each atom is entireied to itself Assumption: The kindred is big-O of is mutual balance F, In other utterance, f(n) E O(f(n)). [5 points] (b) Suffer f(n): Z+ → R+ and g(n) : Z+ → R+. DEFINITION 2. A kindred on a firm is antisymmetric if whenever an atom X is entireied to an atom Y and Y is entireied X, then X = Y. Assumption: The kindred is big-O of is antisymmetric balance F. In other utterance, if f(n) E O(g(n)) and g(n) O(f(n)), then f(n) - g(n). [5 points] Suffer e(n): Z+ → R+, f(n) : Z+ → R+ and g(n) : Z+ → R+. DEFINITION 3. A kindred on a firm is developed if whenever an atom X is entireied to Y and Y is entireied Z, then X is entireied to Z Assumption: The kindred is big-O of is developed balance F. In other utterance, if e(n) E O(f(n)) and f(n) O(g(n)), then e(n) O(g(n)). [5 points] (c) (d) Is is big-O of a inequitable prescribe on F? [5 points] DEFINITION 4. A kindred is a inequitable prescribe on a firm if it is mutual, antisymmetric and developed 8. Affordn an unbounded manage s = Σ f(n), where f(n) is a gentleman balancebearing monotonically moulded creasing exercise that converges and n EZ+, s can be bounded, using unbefitting unimpaireds, as follows k+1 Using the disparity in (1) and 5, assay that Σ E Θ(1). [10 points]

    Coercion each assumption in 7(a)-(c), assay the assumption instantly from the determination of the big-O asymptotic notation if it is gentleman by opinion values coercion the fixeds c and n_0. On the other artisan, if the assumption is bogus, afford a counter-example. Then solution the inquiry in 7(d). F denotes the firm of unimpaired exercises from Z^+ to R^+. (a) Suffer f(n): Z^+ rightarrow R^+. A kindred on a firm is mutual if each atom is entireied to itself. The kindred “is big-O of” is mutual balance F, In other utterance, f(n) atomof O(f(n)). (b) Suffer f(n): Z^+ rightarrow R^+ and g(n): Z^+ rightarrow R^+. A kindred on a firm is antisymmetric if whenever an atom X is entireied to an atom Y and Y is entireied X, then X = Y. The kindred “is big-O of” is antisymmetric balance F. In other utterance, if f(n) atomof O(g(n)) and g(n) atomof O(f(n)), then f(n) = g(n). (c) Suffer e(n): Z^+ rightarrow R^+, f(n): Z^+ rightarrow R^+ and g(n): Z^+ rightarrow R^+. A kindred on a firm is developed if whenever an atom X is entireied to Y and Y is entireied Z, then X is entireied to Z. The kindred “is big-O of” is developed balance F. In other utterance, if e(n) atomof O(f(n)) and f(n) atomof O(g(n)), then e(n) atomof O(g(n)). (d) Is “is big-O of” a inequitable prescribe on F? A kindred is a inequitable prescribe on a firm if it is mutual, antisymmetric and developed. Affordn an unbounded manage s = sigma^infinity_n = 1 f(n), where f(n) is a gentleman balancebearing monotonically decreasing exercise that converges and n atomof Z^+, s can be bounded, using unbefitting unimpaireds, as follows: sigma^k_n = 1 f(n) + unimpaired^infinity_k + 1 f(n) dn lessthanorequalto s lessthanorequalto sigma^k_n = 1 f(n) + unimpaired^infinity_k f(n) dn, k atomof Z^+ Using the disparity in (1) and k = 5, assay that sigma^infinity_n = 1 1/n^3 atomof theta (1).

    Expert Solution

     

    $$int_{5}^{infty} frac{1}{n^3} dn$$ converges to a fixed frac{-1}{2*n^2} is unimpaired of this and then we can harmonious prostrate values and succeed earn a fixed coercion it, suffer this ocnstant be k

    So

    s =sum_{n=1}^5 frac{1}{n^3} + k

    sum_{n=1}^5 frac{1}{n^3} + k

    so

    s = fixed = theta(1)