MATH133 assignment 2 questions (1 Viewer)

[zxl]

yes i'm struggling with the following:

Calculus:
Find the limit of a(n) = n(2^1/n - 1) if exists
and
"Find sequences a(n) and b(n) such that a(n) has a subsequence converging to 2, b(n) has a subsequence converging to 3, and a(n) + b(n) --> 4"

Algebra:
"If H is an m x n matrix and K is an n x m matrix with n < m, prove that HK is not invertible."

anyone got some clues for any?

 

[wogboy]

Find the limit of a(n) = n(2^1/n - 1) if exists

As n approaches what? Infinity? Then try L'Hopital's rule, noting that a(n) = n*(2^(1/n) - 1) = {2^(1/n) - 1}/{1/n}

"Find sequences a(n) and b(n) such that a(n) has a subsequence converging to 2, b(n) has a subsequence converging to 3, and a(n) + b(n) --> 4"

The simplest answer would be to think of a(n) and b(n) both as two simple alternating sequences of the form G*(-1)^n + H, for some numbers G and H. Tailor a(n) and b(n) such that they always add to four, so you have covered the requirement a(n) + b(n) -> 4 as n -> infinity. Remember a(n) and b(n) don't need to converge, but a(n) + b(n) does need to converge (to 4).

The next requirement is to ensure that a(n) has a subsequence converging to 2, and b(n) has a subsequence converging to 3. This is not too hard to do, so I'll let you think about it (remember what a subsequence is)

"If H is an m x n matrix and K is an n x m matrix with n < m, prove that HK is not invertible."

A proof by contradiction is probably the most appropriate here.
suppose HK is invertible:
-> there exists a vector x in R^n such that:
(HK)x = u, for every vector u in R^n
-> there exists x in R^n such that:
H(Kx) = u, for all u in R^n.
-> The function T(v) = Hv is an invertible function. ...(A)

But H is an m x n matrix with n < m,
-> H is not a square matrix
-> T(v) = Hv is not invertible
-> Contradiction with (A) !!
-> HK is not invertible.

 

[zxl]

thx wogboy, how abt this one:
suppose a(n) --> L as n --> infinity, and b(n) = f(a(n)) where b(n) is continuous, prove that f (a(n) ) --> f (L). any thoughts on that as well?

 

[Dash]

question

how do u do this one??

Prove that if An>= 0 for all n and sigma An converges then so does sigma An^2

thx

 

[zxl]

how do u do this one??

Prove that if An>= 0 for all n and sigma An converges then so does sigma An^2

thx

To prove convergence you basically need to show that it is bounded above and increasing (or bounded below and decreasing), in this case, a(n) is obviously incresing since all An positive, and it is smaller than (bounded above by) (An)^2.

 

[actuarial840724]

For C5., how can u prove that sigma An is less than sigma An^2 ??!! Say if An=1/n, then An = (1/n)^2, and u cant prove this question by this way.

 

[zxl]

opps, i mean a^2(n) is positive and increasing, and (a^2 )(n) < a(n)^2. Since a(n)^2 is convergent, so (a^2 )(n) is bounded above

still no clue for the Calculus 1st question tho:
"suppose a(n) --> L as n --> infinity, and b(n) = f(a(n)) where b(n) is continuous, prove that f (a(n) ) --> f (L). "