Questions tagged [calculus]
For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.
138,136 questions
- Bountied 1
- Unanswered
- Frequent
- Score
- Trending
- Week
- Month
- Unanswered (my tags)
1
vote
0
answers
25
views
When is it justified to take a limit inside a series?
Let
$$
f(x)=\sum_{k=1}^{\infty}(-1)^k(k+1),円\chi_{\left(\frac1{k+1},,円\frac1k\right]}(x),
\qquad x\in(0,1].
$$
Thus $f$ is constant on each interval $\left(\frac{1}{k+1},\frac{1}{k}\right],ドル taking ...
1
vote
0
answers
34
views
Find $\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \sum_{n=1}^{\infty} \frac{n + j + i}{n j i (n + j)(n + i)(j + i)}$
how to find the following series:
$$\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \sum_{n=1}^{\infty} \frac{n + j + i}{n j i (n + j)(n + i)(j + i)}$$
what i attempted was using symmetry like this
\begin{...
0
votes
0
answers
45
views
Does the integral $\int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)},円dx$ have a known closed form? [duplicate]
I am studying the definite integral
$$
I = \int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)},円dx .
$$
The integral does converge:
as $x \to 0,ドル $\ln(1+x) \sim x$ and $\ln(1-x) \sim -x,ドル so the ratio tends to $-...
0
votes
0
answers
33
views
Proof that the sum of 2 functions that are k times continuously differentiable results in a function that is also k times continuously differentiable [closed]
I'm starting my linear algebra studies and came across the following statemtent:
E = F(R;R) is the vector space of the one variable real functions $f:\mathbb{R} \rightarrow \mathbb{R}$ . For each k $\...
0
votes
0
answers
56
views
Must $g(4)$ satisfy any inequality if only $g(1),ドル $g'(1),ドル and $g''(1)$ are known?
Consider an everywhere differentiable function $g(x)$. Suppose we are given only the following information at the single point $x=1$:
$g(1) = 7,ドル $g'(1) = -3,ドル $g''(1)>0.5$
We are asked which ...
0
votes
1
answer
76
views
Convergence of $\sum_{n=1}^{\infty} a_n,$ $\text{ where }$ $a_n = \prod_{k=1}^{n} \sin^2(2^k x)$ $\text{ and }$ $x \in (-\infty, +\infty)$ [closed]
To determine the convergence of the series $$\sum_{n=1}^{\infty} a_n, \text{ where } a_n = \sin^2 x \sin^2 2x \dots \sin^2 2^{n}x \text{ and } x \in (-\infty, +\infty).$$
I attempted to use the ratio ...
1
vote
0
answers
67
views
How to extend series coefficient for negative coefficients?
Assume that $f(x)$ is such that there is a $g(x)$ such that $C_k(f(x)) = \frac{1}{k!}C_k(g(x))$ for all $k \geq 0$. It follows that $$\lim_{x \to 0}{\underbrace{\int\ldots\int}_{k \text{ times}} f(x)...
7
votes
9
answers
406
views
Evaluate $\int_0^{\frac\pi2} \frac{1}{a \sin ^2x+b \cos ^2x} ,円 \mathrm dx$
Find the value of
$$\int_0^{\frac\pi2} \frac{1}{a \sin ^2x+b \cos ^2x} ,円 \mathrm dx,$$
where $a,ドル $b>0$.
The corresponding indefinite integral evaluates to
$$\int \frac{1}{a \sin ^2x+b \cos ^2x} \...
3
votes
1
answer
84
views
$f(0)=1,ドル $f(x) \ge 0 \ge f'(x),ドル $f''(x)\le f(x)$ for $x\ge 0$
Problem
Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x),ドル for all $x\ge 0$. Show ...
3
votes
1
answer
63
views
Range of base $a$ such that $f(x) = a^x - bx + e^2$ has two distinct zeros for all $b > 2e^2$
I am trying to solve the following problem involving a function with parameters $a$ and $b$.
The Problem:
Given the function $f(x) = a^x - bx + e^2$ where $a > 1$ and $x \in \mathbb{R}$.
Discuss ...
0
votes
2
answers
91
views
Absolute convergence and rearrangements
I have a few questions regarding the author’s proof of the following theorem:
I don't understand the part where the author claims that absolute value of each term of $t_m - s_N$ is in the tail of the ...
7
votes
0
answers
132
views
how to solve $\mathcal{J}=\int_0^{\pi/2} \frac{x \arcsin(\cos x)}{\sqrt{1 + \sin^2 x} + \cos x} ,円 dx$
what I tried was that $x \in [0, \pi/2]$ meaning $\cos x \in [0,1]$. therefore
$$
\arcsin(\cos x) = \frac{\pi}{2} - x
$$
so the integral is
$$
\mathcal{J} = \int_{0}^{\pi/2} \frac{x\left(\frac{\pi}{2} ...
3
votes
2
answers
85
views
Problem with understanding the technique used to calculate $B_n=\frac{2u_0}{\pi}\int_{x=0}^\pi\sin(nx)(\sin x+\sin 3x)dx$ for heat equation.
This is a problem and answer from my notes:
Solve heat equation for $l=\pi$ and with the initial and boundary conditions:
$U(0,t)=U(\pi,t)=0;\;\;U(x,0)=u_0(\sin x+\sin 3x)$
The answer to the above ...
-3
votes
0
answers
47
views
Divergence of a series based on the sequence [closed]
If a sequence [an] does not converge, that is the limit of "an" as n tends to infinity" does not exist, will the series be referred to as convergent or divergent??
-1
votes
2
answers
57
views
Does uniformly continuous functions apply to something like "sandwich theorem"? [closed]
Suppose $f,g$ are two uniformly continuous functions on $\mathbb R,ドル and $h$ is a continuous function on $\mathbb R$ that satisfies:$$f(x)\le h(x) \le g(x)$$Does that mean $h$ is a uniformly ...