Math Help Forum - Analysis, Topology and Differential Geometry http://www.mathhelpforum.com/math-help en Fri, 20 Nov 2009 23:00:41 GMT vBulletin 60 http://www.mathhelpforum.com/math-help/images/MHF/misc/rss.jpg Math Help Forum - Analysis, Topology and Differential Geometry http://www.mathhelpforum.com/math-help Show that g is differentiable and bounded http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115806-show-g-differentiable-bounded.html Fri, 20 Nov 2009 22:08:09 GMT Given the piecewise function

g(x) =

{ x^(3)e^(-x^(2)/4)sin(4/x^(2)) if x=/=0
{ 0 if x = 0

Prove that g is differentiable and g' is bounded on (-infinity,infinity).

I know that I have to use the definition of a derivative for the first part (not too sure about how to do that). How to show it's bounded, I am not sure.

This is the first time I have ever seen anything like this.

Thanks for any help. ]]> Analysis, Topology and Differential Geometry katielaw http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115806-show-g-differentiable-bounded.html limit and summation help http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115784-limit-summation-help.html Fri, 20 Nov 2009 20:41:28 GMT Prove that the limit as p approaches infinity of the summation from k=1 to infinity of 1/k^p =1 and the limit as p approaches 1+ of the summation from k=1 to infinity of 1/k^p= infinity. Thanks! Prove that the limit as p approaches infinity of the summation from k=1 to infinity of 1/k^p =1 and the limit as p approaches 1+ of the summation from k=1 to infinity of 1/k^p= infinity.

Thanks! ]]> Analysis, Topology and Differential Geometry friday616 http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115784-limit-summation-help.html more summation help http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115782-more-summation-help.html Fri, 20 Nov 2009 20:32:34 GMT Let P and Q be polynomials of degree p and q. Suppose that the coefficient of x^p in P is positive, the coefficient of x^q in Q is positive, and that Q(k) does not equal zero for all positive integers k. Prove that the series summation from k=1 to infinity of P(K)/Q(k) converges if and only if p < q-1.

Thanks for any help! ]]> Analysis, Topology and Differential Geometry friday616 http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115782-more-summation-help.html summation help http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115781-summation-help.html Fri, 20 Nov 2009 20:24:21 GMT Let p be a positive integer. Find the sum of the series: the summation from k=1 to infinity of 1/(k(k+p)). Let p be a positive integer. Find the sum of the series:

the summation from k=1 to infinity of 1/(k(k+p)). ]]> Analysis, Topology and Differential Geometry friday616 http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115781-summation-help.html how to find the inner product formulla http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115774-how-find-inner-product-formulla.html Fri, 20 Nov 2009 19:31:35 GMT once i solve that one is the derivative of the other but here its much harder to guess the formulla http://i45.tinypic.com/14dhhn5.jpg what is the general method? once i solve that one is the derivative of the other

but here its much harder to guess the formulla
http://i45.tinypic.com/14dhhn5.jpg


what is the general method? ]]> Analysis, Topology and Differential Geometry transgalactic http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115774-how-find-inner-product-formulla.html Product of Series http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115728-product-series.html Fri, 20 Nov 2009 12:34:15 GMT Let \sum_{n=0}^{\infty} a_n and \sum_{n=0}^{\infty} b_n be absolutely convergent (complex) series with sums A and B respectively. For each n, define c_n = \sum_{m=0}^{n} a_m b_{n-m}.

1. Show that \sum_{n=0}^{\infty} c_n is absolutely convergent. [Hint: Follow the same basic plan as used in Prop 5.2 (a) ]

Now the proposition says : Suppose \sum_{n=0}^{\infty} a_n is an absolutely convergent series (in \mathbb{C} ) which has sum S. Then any rearrangement is also absolutely convergent and has sum S.

I am stumped I don't even have a beginning of an idea what to do :(. There are even parts after this question too which make even less sense to me but I think tackling this bit is the first step. any help would be appreciated ]]> Analysis, Topology and Differential Geometry slevvio http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115728-product-series.html Real Analysis: Uniform continuity proof http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115704-real-analysis-uniform-continuity-proof.html Fri, 20 Nov 2009 07:48:28 GMT I need some help/advice doing the following proof. Also, I'm in a beginning real analysis class and the section we're covering is on uniform continuity, so keep that in mind I guess.

Suppose f is uniformly continuous on each of the sets X_1, X_2, ..., X_n and let X = \bigcup_{i=1}^{n} X_i. Show that f need not be continuous on X. Show that, even if f is continuous on X, f need not be uniformly continuous on X.

I think it should suffice to show the result for X = X_1 \cup X_2, since you could do a simple induction proof to show the result for up to X_n. Other than that though, I'm confused as to where I should even begin. Any help is greatly appreciated. ]]> Analysis, Topology and Differential Geometry tonyc4l http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115704-real-analysis-uniform-continuity-proof.html sequence http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115694-sequence.html Fri, 20 Nov 2009 04:53:53 GMT 1, X1>sqr(a). X(n+1) = (a+X(n)) / (1+X(n)) Prove that X1>X3>X5.............. and X2 a>1, X1>sqr(a).

X(n+1) = (a+X(n)) / (1+X(n))

Prove that X1>X3>X5..............
and X2<X4<X6..............




Thank you ]]> Analysis, Topology and Differential Geometry felixmcgrady http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115694-sequence.html Multivariable Differential Mapping http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115688-multivariable-differential-mapping.html Fri, 20 Nov 2009 04:09:42 GMT If p:\mathbb{R}^n\to\mathbb{R}^m is a linear map plus some constant and f:A\subset\mathbb{R}^m\to\mathbb{R}^s is k times differentiable, prove that: \textbf{D}^k(f\circ p)(x_0)(x_1,...,x_k)=\textbf{D}^k f(p(x_0))(\textbf{D}p(x_0)(x_1),...,\textbf{D}p(x_0)(x_k)) So I started by induction.... If p:\mathbb{R}^n\to\mathbb{R}^m is a linear map plus some constant and f:A\subset\mathbb{R}^m\to\mathbb{R}^s is k times differentiable, prove that:

\textbf{D}^k(f\circ p)(x_0)(x_1,...,x_k)=\textbf{D}^k f(p(x_0))(\textbf{D}p(x_0)(x_1),...,\textbf{D}p(x_0)(x_k))

So I started by induction. Using the chain rule,

\textbf{D}(f\circ p)(x_0)(x_1)=(\textbf{D}f(p(x_0))\textbf{D}p(x_0))(x_1)

Terrific. So next I assumed that the \textbf{D}^k statement was true. Proceeding for \textbf{D}^{k+1}, I looked at:

\textbf{D}\bigg(\textbf{D}^k(f\circ p)(x_0)(x_1,...,x_{k+1})\bigg)

Here's where I run into trouble though. I would like to use the induction hypothesis on the stuff inside the big parenthesis, but the fact that there are now k+1 components is throwing me off. Another thing that's not helping is the fact that I'm not really comfortable with this notation. I understand that \textbf{D}^k(f\circ p)(x_0) is the kth derivative of f\circ p evaluated at the point x_0, and I'm applying it to (x_1,...,x_k) but is (x_1,...,x_k) a vector, or are x_1,...,x_k each vectors on their own?

So that's where I stand on this problem. Any help is appreciated. ]]> Analysis, Topology and Differential Geometry redsoxfan325 http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115688-multivariable-differential-mapping.html Continuous function of 2 variable http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115669-continuous-function-2-variable.html Fri, 20 Nov 2009 02:39:04 GMT 0, y\in\mathbb{R}\} for \forally_{0} the limitation: \lim_{\substack{x\rightarrow 0^{+}\\y\rightarrow y_{0}}}f(x,y)=\varphi(y_{0}) exists.]]> Assume f(x,y) is continuous on \{(x,y) | x>0, y\in\mathbb{R}\}

for \forally_{0}

the limitation:

\lim_{\substack{x\rightarrow 0^{+}\\y\rightarrow y_{0}}}f(x,y)=\varphi(y_{0})

exists.

now we define function g(x,y) as:

g(x,y) = \begin{cases} f(x,y), & \mbox{if } x>0 \\ \varphi(y), & \mbox{if } x=0 \end{cases}

show that:

g(x,y) is continuous on \{(x,y) | x\geq0, y\in\mathbb{R}\} ]]> Analysis, Topology and Differential Geometry Xingyuan http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115669-continuous-function-2-variable.html need help showing a particular function is a bijection ((URGENT)) http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115660-need-help-showing-particular-function-bijection-urgent.html Fri, 20 Nov 2009 01:47:51 GMT (-1, 1) I know that f(b) = f(a) => b = a... and if you can show this then the function is a bijection. I know I'm forgetting something obvious, but the absolute value in the function is...]]> I need help with the following problem:

Show the function

f(x) = x / (1 + |x|) is a bijection
f: R -> (-1, 1)

I know that f(b) = f(a) => b = a... and if you can show this then the function is a bijection. I know I'm forgetting something obvious, but the absolute value in the function is throwing me for a loop.

Thank you for help you can offer. ]]> Analysis, Topology and Differential Geometry rgriss1 http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115660-need-help-showing-particular-function-bijection-urgent.html Prove fk http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115655-prove-fk.html Fri, 20 Nov 2009 01:09:02 GMT Rm be a segment of diff f as on A converging point wise to f:A-->Rm suppose Dfk are continuous and converge uniformly to g. Then f is differentiable and Df=g]]> Prove that if fk: A in Rn-->Rm be a segment of diff f as on A converging point wise to f:A-->Rm suppose Dfk are continuous and converge uniformly to g. Then f is differentiable and Df=g ]]> Analysis, Topology and Differential Geometry dabien http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115655-prove-fk.html Injective functions http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115653-injective-functions.html Fri, 20 Nov 2009 00:55:51 GMT Let f : A → B et g : B → C be 2 functions

Show that if g ◦ f is injective, then f has to be injective ]]> Analysis, Topology and Differential Geometry hebby http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115653-injective-functions.html finite and infinite sets...denumerable http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115646-finite-infinite-sets-denumerable.html Fri, 20 Nov 2009 00:22:01 GMT Given Q is denumerable, such that R is not denumerable. Show now that R\Q is not denumerable. Given Q is denumerable, such that R is not denumerable. Show now that R\Q is not denumerable. ]]> Analysis, Topology and Differential Geometry hebby http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115646-finite-infinite-sets-denumerable.html Metric Space, closed sets in a closed ball http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115645-metric-space-closed-sets-closed-ball.html Fri, 20 Nov 2009 00:19:17 GMT Let(X, d) be a metric space. the set {y ∈ X : d(x, y) ≤ r} is a closed ball centered at X and with radius r.
(a)Show that a closed ball is a closed set. ]]> Analysis, Topology and Differential Geometry hebby http://www.mathhelpforum.com/math-help/analysis-topology-differential-geometry/115645-metric-space-closed-sets-closed-ball.html