Homework Statement
Determine the complexity of the following code:
for (i = 0; i < 2*n; i += 2)
{
for (j=n; j > i; j--)
{
a++;
}
}
The Attempt at a Solution
Well.. The first for block is O( n ) because i is incremented by 2 each loop up to 2n. The second block...
Homework Statement
For any positive integer n determine:
\sum\limits^n_{i=0} \frac{1}{i!(n-i)!}
Homework Equations
I don't really know where to start.. Up until this point we've just been doing permutations, combinations, and determining the coefficient of a certain term in the expansion of...
Ah, perfect. I love you guys, you're so helpful. I guess I need to be a little more critical of my own work though, before I give up on what I've done completely.
Now though, knowing A & B (I guess I was solving for them wrong), I am still confused as to how to get the 'nth term' of a series...
Alright, well this is what I've done:
\frac{6}{n(n+1)} = \frac{A}{n} + \frac{B}{n+1}
\frac{6}{n(n+1)} = \frac{A(n+1) + Bn}{n(n+1)}
6 = A(n+1) + Bn
6 = An + A + Bn
Split into two equations,
1: 6 = A + B
2: 6 = A
Then,
2->1: 6 = 6 + B
B = 0
I can see I'm doing something wrong, because...
Alright, I've separated the equation into the following:
\frac{6}{n(n+1)} = \frac{A}{n} + \frac{B}{n+1}
And solved for A, B, getting A = 6, B = 0.
So, I'm left with:
\sum\limits^\infty_{n=1} \frac{6}{n}
Now, I'm lost as to what to do. I don't know how to solve for this sum...
Any hints...
Homework Statement
Evaluate the sum of the following:
\sum\limits^\inf_{n=1} \frac{6}{n(n+1)}
Homework Equations
The Attempt at a Solution
Well... The denominator is going to get infinitely large as n approaches infinity, so would the value of the sum not converge to zero? The...
Homework Statement
Does the following interval diverge?
\int^9_1 \frac{-4}{\sqrt[3]{x-9}}
Homework Equations
The Attempt at a Solution
Well.. I've used the following function that I think is always less than the above function to prove that the function above DOES NOT diverge (by...
Homework Statement
\int^1_0 \frac{-3x^3+12x+30}{x^2-x-6}dx
Homework Equations
The Attempt at a Solution
I've attempted long division, but the long division does not seem to come to an end.. I'm not sure what to make of this.. If the long division does not end, does this mean that...
Integral (Partial-Frac Decomp) **SIMPLE?
Homework Statement
\int^3_2 \frac{-dx}{x^2-1}
Homework Equations
The Attempt at a Solution
= \int^3_2 \frac{A}{x} + \frac{B}{x-1} dx For some integers A and B.
-1 = A(x-1) + B(x+0)
-1 = Ax - A + Bx
Split into two equations...
Homework Statement
I missed one class on trigonometric identities in integrals, and I feel that one is needed here:
\int^3_0\frac{1+arctan(\frac{x}{3})}{9+x^2}dx
Homework Equations
The Attempt at a Solution
Again, I'm unsure what to do. I think that it is a trig identity, but I could be...
Homework Statement
Find the derivative of the function
F(x) = \int^0_{x^2-1}\frac{sin(t+1)}{t+1}dt
Homework Equations
The Attempt at a Solution
F'(x) = -\frac{sin(x^2)}{x^2}
I'm just learning this and unsure if this is correct. It seems too easy?
Homework Statement
\int^2_1 6x\sqrt{x-1}dx
Homework Equations
The Attempt at a Solution
Let u=x-1.
Then, u+1=x,
and du=dx.
Continued from problem statement,
=6 \int^1_0 (u+1)u^{\frac{1}{2}}du
=6 \int^1_0 u^{\frac{3}{2}} + u^{\frac{1}{2}}du
=6(1^{\frac{3}{2}} + 1^{\frac{1}{2}})
=6(2)
=12...
Homework Statement
\int \frac{-2x}{\sqrt[4]{x+2}}dx
The Attempt at a Solution
=-2*\int x(x+2)^{\frac{1}{4}}dx
Let u=x+2.
Then, u-2=x,
and du = dx
.. Continued from above,
=-2*\int (u-2)u^{\frac{1}{4}}du
=-2*\int u^{5/4}-2u^{\frac{1}{4}}du
Is that last step allowed?
Homework Statement
Calculate the following:
\int^{-4}_{-6} (x^-1+5x)dx
Homework Equations
The Attempt at a Solution
I've worked this down to
ln(-4) + 5(-4) - ln(-6) - 5(-6)
=ln(-4)-ln(-6)+10
The answer to the left half (the ln parts) of the equation is undefined, and the...
Homework Statement
How many was can 3 identical prizes be awarded to 98 potential winners?
Homework Equations
The Attempt at a Solution
Well. I know that if the prizes were unique, the first prize would have 98 possible winners, the second prize would have 97 possible winners...
If anyone has anything close to an idea of what this could mean.. Please help. I just need a good guess so I can try the question, but I don't have a clue.
Homework Statement
Let Z be the set of all integers.
Then, S is a relation on the set Z x Z defined by:
for (a1, a2), (b1, b2) belong to Z x Z,
(a1, a2)S(b1, b2) <-> a1b2 = a2b1.
Homework Equations
The Attempt at a Solution
The actual problem is about symmetry...
Homework Statement
What is the integral of x(e^x)?
The Attempt at a Solution
It's part of a larger question. I've got a midterm tomorrow and just realized I don't know the principles of integrating e^x other than that the integral of e^x is e^x. I've scoured my textbook and cannot find...
Homework Statement
Determine whether or not the integral from 0 to 1 of (5ln(x)) / ( x^(3/2) ) converges or not.
Homework Equations
The Attempt at a Solution
I just need to know which end of the integral they are talking about. As x=>0, y=>-infinity. As x=>1, y=>1. I'm assuming...
Homework Statement
Determine whether or not the improper integral from 0 to infinite of (e^x)/[(e^2x)+4] converges and if it does, find it's definite value.
Homework Equations
The Attempt at a Solution
I missed the lecture on the Comparison Test, so I'm essentially useless.
I...
So. I've been told by my prof that the best way to find the determinant of a matrix is to row reduce it to upper triangular and then take the product of the numbers on the diagonal. That's fine, BUT, how do you know how to reduce it?
Depending on what row operations you do, you get...