Homework Statement
\int \frac{4x^5-1}{(x^5+x+1)^2} dx = ?
Homework Equations
The solution is
- \frac{x}{x^5 + x + 1}
The Attempt at a Solution
Other than getting lucky and noticing immediately that this could be the derivative of a fraction, I do not see an easy way to solve this. The...
Homework Statement
Show that if:
lim_{k\to\infty}b_k\to+\infty,
\sum_{k=1}^\infty a_k converges and,
\sum_{k=1}^\infty a_k b_k converges, then
lim_{m\to\infty} b_m \sum_{k=m}^\infty a_k = 0
Homework Equations
The Attempt at a Solution
I only have an idea why this is true--\sum a_k...
In lecture in my real analysis course the other day we were proving that absolute convergence of a series implies convergence. Our professor started off by showing us the wrong way to prove it:
\left| \sum_{k=1}^\infty a_k \right| \leq \sum_{k=1}^\infty \left| a_k \right| < \epsilon
Then he...
I am picking courses for next semester and am torn between a numerical analysis class or one on Fourier series and wavelets. The n.a. course fits my schedule way better but every book and syllabus I have browsed concerning numerical analysis has seemed uninteresting--more like a branch of CS...
I never took complex analysis in undergrad and always regretted it, so I'm working through the book Visual Complex Analysis on my own. Really enjoying it so far.
Homework Statement
Actually you can view the problem...
I apologize for being so dense, but I'm still confused. A couple different books I have print the result
\frac{1}{2}\oint_C -ydx + xdy = \iint_{R} dA = A
If the area of the ellipse is A=\pi a b then I would think that the value of the line integral is 2A.
Homework Statement
Let C be the ellipse with center (0,0), major axis of length 2a, and minor axis of length 2b. Evaluate \oint_C xdy - ydx.Homework Equations
I solved this two ways. First I parameterized x and y as x=a \cos \theta and similarly for y. I also applied Green's theorem, which...
Problem
Suppose for all subgroups H,K of a finite group G, either H \subset K or K \subset H. Show that G is cyclic and its order is the power of a prime.
Attempt
I think I get the intuition: if H and K are not the same, then one of them must be the trivial subgroup and the other must be G...
I'm thinking about taking the math GRE in December but I've never studied abstract algebra--all this about rings and groups just flies right over my head. Can anyone recommend a good introductory book? I'm thinking one of the Dover works might be good since they seem to emphasize problem...
#34 on the much-discussed http://ftp.ets.org/pub/gre/Math.pdf" :
Suppose f is a differentiable function with \lim\limits_{x \to \infty }f(x)=K and \lim\limits_{x \to \infty }f'(x)=L for some K,L finite. Which must be true?
L=0
\lim\limits_{x \to \infty }f''(x)=0
K=L
f is constant.
f' is...