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?