Find the sum of the following infinite series   $$\frac {1}{3!} + \frac {4}{4!} + \frac {9}{5!} + \ldots$$  I was trying this question ,but I couldn't get it. This is homework that my teacher has given me.  I solved this question using d'Alembert's ratio test but I didn't get a finite value.  If anyone could help me, that would be much appreciated. Thank you
Problem is to find sum  $$\frac{1}{3!}+\frac{4}{4!}+\frac{9}{5!}\cdots$$  What I knew doesn't apply on this problem  Some series are telescoping, some types are solvable using binomial , both look useless here Binomial gives  $$n (n-1) (n-2)x^3=1$$  $$n (n-1)( n-2)(n-3)x^4=4$$  What approach to use here?
If someone else is booking international ticket for me using his credit or debit card, does he need to be present physically at the time of check-in?   Suppose I am coming back from Singapore to my home country and my ticket is booked by my friend for me, who lives in my home country. Do airlines in Singapore ask for the person to be present physically to check in? Let $f$ be differentiable at $x=a$.   Prove that if $x_n \to a^+$ and $y_n \to a^-$ then:  $$\lim_{n\to \infty} \frac{f(x_n)-f(y_n)}{x_n-y_n}=f'(a).$$  Every option that I think about seems to my very trivial, so I believe that I am doing something wrong.  Both numerator and denominator approach zero as $n\to\infty$ as the case of the formal definition of derivative, but it isn't guaranteed that the limits are equal (“$\frac{0}{0}$”).   Any direction? How can I get fireworks? What are the requirements for making fireworks appear at the end of a level in the original Super Mario Bros.? How do I change the cursor and its size? In the current era of cricket, we see batsmen using switch hits to play a ball. That is, while playing a ball they change their batting side (left to right handed and vice versa) just before the ball comes to them.  My question is, can a bowler bowl a switch ball? Meaning, can a left hand bowler bowl right handed to a batsman? If it is not permitted, then is it not unfair to the bowlers that batsmen can change their batting style but not the bowlers? Or are there are any rules against a switch hit? Is the Higgs boson an elementary particle? If so, why does it decay? The  is an excitation of the Higgs field and is very massive and short lived. It also interacts with the Higgs field and thus is able to experience mass.   Why does it decay if it is supposed to be an elementary particle according to the standard model? How to pass parameters to an alias? I lost my .keystore file? How did humans and vulcans communicate in Star Trek: First Contact? Setting: Let $\sum a_n z^n$ have radius of convergence $R$.  We have that  $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$  via Hadamard's formula for the radius of convergence.  Question: What are the radii of convergence for (i) $\sum a_n z^{2n}$ and (ii) $\sum a_n^2 z^n$?  Attempt for $\sum a_n^2 z^n$:   For ease of notation, let $R_1$ and $R_2$ denote the radii of convergence for power series $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ respectively. From Hadamard's formula, we then have  $$ R_2 = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n^2 \right|^{1/n}} = \left( \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} \right)^2 = R^2 $$   But what about $R_1$? How to add “jQuery” onclick functionality for a column each row in jQgrid to display different popups for different rows Fatou's lemma and measurable sets I don't know how can I imply Fatou's lemma for any measurable sets $A_k$, that is, $$\lambda(\liminf A_k)\le\liminf\lambda(A_k).$$ How can I prove it? And is there any example in $\mathbb R$ of sequence of measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim\lambda(A_k)=1$, but $\liminf A_k=\varnothing$ ? Thanks for your help! How to combine two jQuery results How do you combine two jQuery search results? eg:  var $allFoos = $('.foo'),     $allBars = $('.bar')     $allFoosAndBars = $allFoos + $allBars;   Obviously, I just made up that last line, but I hope it makes it sorta clear what I mean. To be clear, the example is greatly simplified, and it could be any arbitrary sets i'm talking about, so $('.foo, .bar') is not what I'm after. I am not sure what caused this error, but here is what the whole error says, and also this is sitting as a notification, and preventing me from updating any software using Update Manager - Please provide some assistance or tell me how to figure out what to do to fix it.  Could not calculate the upgrade  An unresolvable problem occurred while calculating the upgrade.  Please report this bug against the 'update-manager' package and include the following error message: 'E:Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.' How to add fonts in LibreOffice writer I want to add nepali fonts like kantipur in Ubuntu 12.04.How can I do this. Consider the space of continuously differentiable functions, $$C^1([a,b]) = \{f:[a,b]\rightarrow \mathbb{R}\mid f,f' \text{ are continuous}\}$$ with the $C^1$-norm $$\lVert f\rVert := \sup_{a\leq x\leq b}|f(x)|+\sup_{a\leq x\leq b}|f'(x)|.$$ Prove that $C^1([a,b])$ is a Banach Space.   This was the proof we were given: Assuming $C^1([a,b])$ is a normed linear space all we need to show is completeness. Let $(f_n)$ be a Cauchy Sequence in $C^1([a,b])$ with respect to the $C^1$-norm. Then each $f_n,f'_n\in (C([a,b]),\|\cdot\|_{\sup})$. We know that $C([a,b])$ is complete and thus there exists $f,g\in C([a,b])$ such that $f_n\rightarrow f$, and $f'_n\rightarrow g$ (uniformly) with respect to $\|\cdot\|_{\sup}$. If we let  $$ F_n(x) = \int_a^x f_n(t)dt, \hspace{2mm}  F(x) = \int_a^x f(t)dt $$ then $F_n\rightarrow F$ uniformly because  $$\lVert F_n-F\rVert_{\sup}\leq \sup_{a\leq x\leq b}\int_a^x|f_n(t)-f(t)|dt\leq \lVert f_n-f\rVert_{\sup}&lt;\epsilon.$$ From the fundamental theorem of calculus: $$f_n(x)-f_n(a) = \int_a^x f'_n(t)dt $$ Since $f'_n\rightarrow g$ uniformly then  $$ \int_a^xf'_n(t)dt\rightarrow \int_a^x g(t)dt $$ Since we know that $f_n\rightarrow f$ uniformly, $$f(x)-f(a) = \int_a^x g(t) dt $$ which by the fundamental theorem of calculues implies $f'=g$. So we know have $f_n\rightarrow f$ and $f'_n\rightarrow g=f'$ which mean $f_n\rightarrow f\in C^1([a,b])$ with respect to $C^1$-norm. So every cauchy sequence converges. Hence $C^1([a,b])$ is a Banach Space.   So I understand most of the proof. Where I get confused is that how did we actually show this satisfies the $C^1$-norm? Maybe I don't understand what this norm actually does.   Thank you for any help, comments and advice! Newton's third law does not apply to the kinetic energy formula? Here is a classic problem:     A 60kg boy and a 40kg girl are sitting in front of each other on skateboards on a frictionless horizontal surface. The boy pushes the girl and the girl gains a velocity of 0.3m/s during the 0.5s the boys hand was in contact with her. What is the boy's resulting velocity?   Considering Newton's Third Law, the best way is to find the total force applied and rearrange the formula to find the boy's speed.  Newton's Law Method:  $F = m*a$  $a=\frac{0.3m/s}{0.5s}$  $F = 40kg * 0.6m/s^2$  $F = 24N$  Therefore the boy's speed will be:   $\frac{24N}{60kg} = a$  $a = 0.4m/s^2$  $v = 0.4m/s^2 * 0.5s$  $v = 0.2m/s$  Kinetic Energy Method:  $E_{k}=\frac{1}{2}*m*v^2$ the girl's kinetic energy  $E_{k}=\frac{1}{2}*40kg*(0.3m/s)^2$  $E_{k}=1.8J$  Since every action has an equal and opposite reaction the boy should have the same amount of kinetic energy.  $E_{k}=\frac{1}{2}*60kg*(0.2m/s)^2$  $E_{k}=1.2J$  As you can see, using Newton's Third Law, the kinetic energy was not the same. So what's going on here? I thought that in this case they should both have the same amount of kinetic energy due to Newton's Law. I must be missing something since there is more energy going in one direction than the other after meeting. My book used the first method as the answer and, well, it feels wrong. Unknown Column In Where Clause I have a simple query:  SELECT u_name AS user_name FROM users WHERE user_name = "john";   I get Unknown Column 'user_name' in where clause.  Can I not refer to 'user_name' in other parts of the statement even after select 'u_name as user_name'? Where to set proxy ignore list? I upgraded to ubuntu 11.10 from 11.04, shocked to see few things...    My system is behind a proxy, so I did set a proxy setting. Being a web developer, I want to remove a lot of addresses that I type in browser address bar to not hit proxy. There's no such ignore list in ubuntu 11.10, what/where do I do it? What was the requirement to change the way network gui worked? Prove that $g$ is tangent to the graph of $f$ So I had these two functions and the following exercise:  $f(x)=x^3-2x^2$ $g_p(x)=px$ Prove that $g_{-1}$ is a tangent line to the graph of $f$. How many points do $g_{-1}$ and $f(x)$ have in common?  So yeah, the points in common are fairly simple to figure out:  Points in common: $x=1$ and $x=0$  So to see if it's actually tangent to the graph, I took the derivative of both functions at the points where they meet. The result:  $f'(x)=3x^2-4x$ $g'_{-1}(x)=-1$  So $f'(1)=-1$ and $f'(0)=0$ So I'm not sure why I get this answer? By the question, we are supposed to prove that its tangent to the graph of $f$ but by my calculation, it seems like only $f(1)$ is tangent with $g_{-1}(1)$. Help is appreciated! (Please avoid hint answers).