# Tagged Questions

For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

218 views

### Is the following Alternating Series Absolutely Convergent?

$$\sum_{n=1}^\infty\frac{(-1)^n}{2n+1}$$ I think it is Absolutely Convergent because it converges by direct comparison to Harmonic series? Am I right or wrong?
120 views

### Finding $\sum\limits_{k=0}^n k$ using summation by parts

This is another exercise from Smoryński's Logical Number Theory; not being a mathematician, I'm a bit new to this finite difference stuff, so, please, bear with me! In a previous exercise, Smoryński ...
90 views

43 views

### how to solve this formula limits's formula [duplicate]

I have already known one way to solve this formula but I just want to know the easier way to do so: $$\lim_{ u \to 0} \frac{\sin(u)}{u}=1$$ Please kindly help me!! Thank You.
40 views

### Evaluating a function at a point where $x =$ matrix.

Given $A=\left( \begin{array} {lcr} 1 & -1\\ 2 & 3 \end{array} \right)$ and $f(x)=x^2-3x+3$ calculate $f(A)$. I tried to consider the constant $3$ as $3$ times the identity matrix ($3I$) but ...
74 views

### What did I do wrong?

So, I have found the following problem. This problem is a multiple-choice one, and I have to pick the correct answer. The problem, gives a function $f:D \to R$, $$f(x)=\frac{xe^x}{e^x-a}$$ with $a$ ...
83 views

47 views

### Proving that the Gamma function $\Gamma(y)$ converges for $y>0$.

How can I justify that $$\Gamma(y)=\int_0^\infty t^{y-1}e^{-t} \, \mathrm{d}t$$ exists for all $y>0$? I'm struggling to compare it to a known convergent integral.
259 views

### Every step function is a linear combination of elementary step functions.

If $J$ is any subinterval of $[a, b]$ and if $\phi_J (x) := 1$ for $x \in J$ and $\phi_J (x) := 0$ elsewhere on $[a, b]$, we say that $\phi_J$ is an elementary step function on $[a, b]$. Then to ...
41 views

### All surfaces through a common “concur-line” [closed]

Find all second degree surfaces passing through a common given parameterized space curve of intersection: $$(x,y,z) = (\pm \sqrt {2 t ( 1-t)} , t , (1-t) )$$ using a single variable parameter ...
30 views

41 views

47 views

I'm asked to find the derivative of the following: $$\sqrt[4]{x} + \sqrt[3]{3x}$$ I attempted to solve the problem and got the following result, but my book says I am wrong. $$\frac 14x^{-\frac ... 5answers 142 views ### A limit problem: \lim\limits_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} } I need help in solving the limit below:$$\lim_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$$What I've done is to simplify ... 1answer 90 views ### Zeros of an analytic function [duplicate] How to prove zeros of a real analytic function (non-zero function) is always countable? 1answer 32 views ### Infinite sub-sequences that make up a sequence A sequence \{a_n\} can be broken into sub-sequences, \{a_n\}^1_{k_1}, \{a_n\}^2_{k_2}, \dots,\{a_n\}^m_{k_m}, if every element in \{a_n\} belongs to at least one of the sub-sequences. I had to ... 4answers 61 views ### Find y' and y'' :  y=x^2\ln(2x) for x> 0 :  y=x^2\ln(2x) Product rule:$$(x^2)\cdot[\ln (2x)]'+ (\ln (2x))\cdot[x^2]' y'= x^2\frac{1}{2x}\cdot (2)+\ln(2x)\cdot(2x) =2x\ln(2x)y''=(2x)[\ln(2x)]'+(\ln(2x))[2x]'+[x]'\$...
I want to calculate $$\lim_{t \to 0} \frac{t^2}{\sin^2(t)}$$ and I proceed as follows $$\stackrel{H}{=} \lim_{t \to 0} \frac{2t}{2\sin(t)\cos(t)} \implies \lim_{t \to 0} \frac{2t}{\sin(2t)}$$ ...