In: Computer Science

1- Show that (n^3+3n^2+3n+1) / (n+1) is O (n2 ). Use the definition and proof of big-O notation.

2- Prove using the definition of Omega notation that either 8 n is Ω (5 n ) or not.

please help be with both

1) The first equation can be easily evaluated to simplified form :

Now,

If , then

Which means for all values of n, the upper range of the given function would have an upper bound of 4n^2.

So, the complexity of the function in Big - Oh Notation would be O(n^2).

2) **Omega Notation**,
**Ω**

The notation is the best way to represent the lower bound of any algorithms running time in terms of the input. It gives an estimate of best time complexity of the function.

For all values of n i.e.
, the relation
holds true. Hence, the function g(n) = 5^n gives a lower bound to
the function f(n) = 8^n for all
.

Hence f(n) can be represented in omega notation as
.

**Please upvote the answer if it helps you.**

Calculate the Big-O time complexity. Show work
1. n^2 + 3n + 2
2. (n^2 + n)(n ^2 + π/2 )
3. 1 + 2 + 3 + · · · + n − 1 + n

6a. Show that 2/n = 1/3n + 5/3n and use this identity to obtain
the unit fraction decompositions of 2/25 , 2/65 , and 2/85 as given
in the 2/n table in the Rhind Mathematical Papyrus.
6b. Show that 2/mn = 1/ (m ((m+n)/ 2 )) + 1/ (n ((m+n)/ 2 )) and
use this identity to obtain the unit fraction decompositions of 2/7
, 2/35 , and 2/91 as given in the 2/n table in the Rhind
Mathematical Papyrus....

Justify the following statements using the "big-Oh"
definition:
a) (n+25)2 is O(n2) , and n2 is
O((n+25)2)
b) n3 is NOT O(n2);
c) Given f1(n) is (n+25)2 , and f2(n) is
n3 what is the big-Oh for f1(n) x f2(n)?

Show that (1 + 2 +. . .+n)2 > 12 +. .
.+ n2, for n ≥ 2.

If f(n) = 3n+2 and g(n) = n, then Prove that f(n) = O (g(n))

Write a combinatorial proof for 1 n + 2 ( n − 1 ) + 3 ( n − 2 )
+ ⋯ + ( n − 1 ) 2 + n 1 = ( n + 2 choose 3 ) .

(a) Find the limit of {(1/(n^(3/2)))-(3/n)+2} and use an
epsilon, N argument to show that this is indeed the correct
limit.
(b) Use an epsilon, N argument to show that {1/(n^(1/2))}
converges to 0.
(c) Let k be a positive integer. Use an epsilon, N argument to
show that {a/(n^(1/k))} converges to 0.
(d) Show that if {Xn} converges to x, then the sequence {Xn^3}
converges to x^3. This has to be an epsilon, N argument [Hint: Use
the difference...

Derive the Sackur-Tetrode equation starting from the
multiplicity givenin Ch. 2:
Ω =(1/N!)(V^{N}/h^{3N})(pi^{3N/2}/3N^{2}!)(2mU)^{3N/2}
The Sackur-Tetrode equation is:
S=Nk[ln((V/N)((4pi*m*U)/(3Nh^{2}))^{3/2})+(5/2)]

Argue with proof that whether following asymptotic notations are
transitive, reflexive, or symmetric.
O(n); o(n); Ω(n); ω(n); ϴ(n)

Show that
(a)Sn=<(1 2),(1 3),……(1 n)>.
(b)Sn=<(1 2),(2 3),……(n-1 n)>
(c)Sn=<(1 2),(1 2 …… n-1 n)>

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