This is the question: Find the sum of $$\frac1 9+\frac2{9^2}+\frac3{9^3}+\ldots +\frac n{9^n}+\ldots$$ What I did was convert the whole equation to $$\frac{1\cdot 9^{n-1}+2\cdot 9^{n-2}+3\cdot 9^{n-3}+\ldots +n}{9^n}$$ I have no clue about what to do next. I would probably think that you would have to identify the sequence but I don't know how to get there from this.
How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
Why are my render times 30+ minutes, but other people's are ~3 minutes? Given the following problem     Let $U$ and $V$ be finite dimensional vector spaces, such that $\dim(U) = \dim(V)$. Prove that $U$ and $V$ are isomorphic.   I was just wondering if somebody could critique and validate my proof.  Proof  Two vector spaces $U$ and $V$ are isomorphic if and only if there exists a bijection $\phi: U \to V$. That is, an invertible linear map between the two. Let $\dim(U) = \dim(V) = n$ and let $\left\{u_1, \dots, u_n\right\}$ and $\left\{v_1, \dots, v_n\right\}$ be bases for $U$ and $V$, respectively. It follows that any vector $u \in U$ can be written as  $$ u = c_1u_1 + \cdots + c_nu_n $$  We define $\phi: U \to V$ as  $$ \phi(c_1u_1 + \cdots + c_nu_n) = c_1v_1 + \cdots + c_nv_n $$  Observe that $\phi$ is invertible. That is, given a vector  $$ v = d_1v_1 + \cdots + d_nv_n $$  in $V$, we define $\phi^{-1}: V \to U$ as  $$ \phi^{-1}(d_1v_1 + \cdots + d_nv_n) = d_1u_1 + \cdots + d_nu_n $$  Since there exists a bijection between $U$ and $V$, they are isomorphic. The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ How to compute this limit:  $$\lim_{n\to\infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+n^2}\right)$$  Please give me some hint. Are ores replenished over time? Applications of ultrafilters I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters?  I'm at the beginning, so I'd prefer some applications that also a beginner could read. checking if a number is a prime I was reading Wikipedia, and it was given that "all primes are of the form 6k ± 1"  (other than 2 and 3), where k = 1,2,3,4,...  Is this statement correct? If yes, can we use this to test if a given number is a prime number? For instance, we can say that 41 is a prime number, since there exists an integer K (K >= 1), such that 6k - 1 = 41 ==> k = 7.  I am confused why this test cannot be used to test if a number is prime?  Thanks, Sekhar Suppose I am given a $2$x$2$ matrix $A=$ \begin{pmatrix} a &amp; b \\ c &amp; d \end{pmatrix}  And an initial vector $x_n$ = \begin{pmatrix} x_0 \\ y_0\end{pmatrix}.  Under repeated iteration $x_{n+1} = Ax_n$, we find that $x_{n+1}$ converges to a line in the direction of the $eigenvector$  of $A$ of the form $X$= $[u,mu]$ where $m$ is the slope of the line.  I know that this process somehow also involves the largest $eigenvalue, \lambda,$ of $A$, though I am unsure how it connects.  My question is: How does the largest eigenvalue connect to this repeated iteration, how do I find the slope of the line of convergence $m$, and what is meant by the "direction of the eigenvector"? When reading a stack trace like:  [FormatException: Input string was not in a correct format.]    System.Number.StringToNumber(String str, NumberStyles options, NumberBuffer&amp; number, NumberFormatInfo info, Boolean parseDecimal) +2755599    System.Number.ParseInt32(String s, NumberStyles style, NumberFormatInfo info) +112    System.Convert.ToInt32(String value) +68   What do the +68, +112, etc. numbers mean. I'm guessing that they're offsets to something. If so, what? In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance. How are qubits better than classical bit if they collapse to a classical state after measurement? Classical computers store information in bits, which can either be $0$ or $1$, but, in a quantum computer, the qubit can store $0$, $1$ or a state that is the superposition of these two states. Now, when we make a "measurement" (to determine the state of qubit), it changes the state of the qubit, collapsing it from its superposition of $0$ and $1$ to the specific state consistent with the measurement result.  If from a measurement of a qubit we obtain only a single bit of information ($0$ or $1$, because the qubit collapses to either of the states), then, how is a qubit better than a classical bit? Since ultimately we are only able to get either $0$ or $1$ from either classical or quantum bit.  If you can explain with an example, it may be helpful. Why would I open a low-rate 9 month CD instead of a higher rate savings account? One specific (popular online) bank is offering a 9 month 1.25% CD, while it's savings account is 1.90%.  There's no way I see interest rates tumbling in the next nine months, so why should I -- why should anyone -- get a 1.25% CD when savings accounts are paying 52% more interest?  (This question is not specific to this particular bank at this particular time.  It's about the general practice of buying low-rate CDs when savings accounts have a higher rate.) How to Pass an Object Method as a Parameter in Delphi, and then Call It? I fear this is probably a bit of a dummy question, but it has me pretty stumped.  I'm looking for the simplest way possible to pass a method of an object into a procedure, so that the procedure can call the object's method (e.g. after a timeout, or maybe in a different thread).  So basically I want to:   Capture a reference to an object's method. Pass that reference to a procedure. Using that reference, call the object's method from the procedure.   I figure I could achieve the same effect using interfaces, but I thought there was another way, since this "procedure of object" type declaration exists.  The following doesn't work, but might it help explain where I'm confused...?  interface    TCallbackMethod = procedure of object;    TCallbackObject = class     procedure CallbackMethodImpl;     procedure SetupCallback;   end;  implementation  procedure CallbackTheCallback(const callbackMethod: TCallbackMethod); begin   callbackMethod(); end;  procedure TCallbackObject.CallbackMethodImpl; begin   // Do whatever. end;  procedure TCallbackObject.SetupCallback; begin   // following line doesn't compile - it fails with "E2036 Variable required"   CallbackTheCallback(@self.CallbackMethodImpl); end;   (Once the question is answered I'll remove the above code unless it aids the explanation somehow.) Proof of triangle inequality I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :) Deny access to a webpage using web.config How to get a function name as a string? In Python, how do I get a function name as a string, without calling the function?  def my_function():     pass  print get_function_name_as_string(my_function) # my_function is not in quotes   should output "my_function".  Is such function available in Python? If not, any ideas on how to implement get_function_name_as_string, in Python? C# - how can I get owner's name for a Mutex I have a shared mutex between 2 applications. I want to create an administration console that shows the current owner's name of the mutex.  How can I get the application name that currently owns the Mutex? Find $P(12)$ when $P(x)={1\over(x+1)}$, for $x=0,1,2,......11$ let $P(x)$ be a polynomial of degree $11$ such that  $P(x)={1\over(x+1)}$, for $x=0,1,2,......11$  then find value of $P(12)$ In The Winter Soldier, who did Peggy Carter say she married? When Steve Rogers visits Peggy Carter, she has a family photo on her nightstand and says that she married someone after Cap disappeared. Who did she marry? Linear regression not fitting well I do a linear regression using R lm function:  x = log(errors) plot(x,y) lm.result = lm(formula = y ~ x) abline(lm.result, col="blue") # showing the "fit" in blue     but it does not fit well. Unfortunately I can't make sense of the manual.  Can someone point me in the right direction to fit this better?  By fitting I mean I want to minimize the Root Mean Squared Error (RMSE).    Edit: I have posted a related question (it's the same problem) here:   and the raw data here:    except that on that  x is what is called errors on the present page here, and there are less samples (1000 vs 3000 in the present page plot). I wanted to make things simpler in the other question. How does the Commander interact with Banishing Light or Oblivion Ring?