Questions tagged [calculus]

Question 1

Let $$ f(x)=\sum_{k=1}^{\infty}(-1)^k(k+1)\,\chi_{\left(\frac1{k+1},\,\frac1k\right]}(x), \qquad x\in(0,1]. $$ Thus $f$ is constant on each interval $\left(\frac{1}{k+1},\frac{1}{k}\right]$, taking ...

Question 2

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{...

Question 3

I am studying the definite integral $$ I = \int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx . $$ The integral does converge: as $x \to 0$, $\ln(1+x) \sim x$ and $\ln(1-x) \sim -x$, so the ratio tends to $-...

Question 4

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 $\...

Question 5

Consider an everywhere differentiable function $g(x)$. Suppose we are given only the following information at the single point $x=1$: $g(1) = 7$, $g'(1) = -3$, $g''(1)>0.5$ We are asked which ...

Question 6

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 ...

Question 7

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)...

Question 8

Find the value of $$\int_0^{\frac\pi2} \frac{1}{a \sin ^2x+b \cos ^2x} \, \mathrm dx,$$ where $a$, $b>0$. The corresponding indefinite integral evaluates to $$\int \frac{1}{a \sin ^2x+b \cos ^2x} \...

Question 9

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 $f''(x) \le f(x)$, for all $x\ge 0$. Show ...

Question 10

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 ...

Question 11

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 ...

Question 12

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} ...

Question 13

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 ...

Question 14

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??

Question 15

Suppose $f,g$ are two uniformly continuous functions on $\mathbb R$, 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 ...

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Questions tagged [calculus]

When is it justified to take a limit inside a series?

Find $\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \sum_{n=1}^{\infty} \frac{n + j + i}{n j i (n + j)(n + i)(j + i)}$

Does the integral $\int_{0}^{1} \frac{\ln(1+x)}{\ln(1-x)}\,dx$ have a known closed form? [duplicate]

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]

Must $g(4)$ satisfy any inequality if only $g(1)$, $g'(1)$, and $g''(1)$ are known?

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]

How to extend series coefficient for negative coefficients?

Evaluate $\int_0^{\frac\pi2} \frac{1}{a \sin ^2x+b \cos ^2x} \, \mathrm dx$

$f(0)=1$, $f(x) \ge 0 \ge f'(x)$, $f''(x)\le f(x)$ for $x\ge 0$

Range of base $a$ such that $f(x) = a^x - bx + e^2$ has two distinct zeros for all $b > 2e^2$

Absolute convergence and rearrangements

how to solve $\mathcal{J}=\int_0^{\pi/2} \frac{x \arcsin(\cos x)}{\sqrt{1 + \sin^2 x} + \cos x} \, dx$

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.

Divergence of a series based on the sequence [closed]

Does uniformly continuous functions apply to something like "sandwich theorem"? [closed]

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