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convergency

convergency (kənˈvɜːdʒənsɪ) [f. as prec. + -ency.] 1. The state or quality of being convergent.1709 Berkeley Th. Vision §35 The convergency or divergency of the rays. 1831 Brewster Optics iv. §41 Rays of different degrees of divergency and convergency. 1846 Joyce Sci. Dial. xvii. 312 To collect the ... Oxford English Dictionary

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Convergency of $\sum_{n=1}^{\infty}\frac{\csc(n)}{n!}$ I am stuck on how to prove the convergency of the series $$\sum_{n=1}^{\infty}\frac{\csc(n)}{n!}.$$ It seems like that the series converges to approximately $2.85...

$\pi\not\in\mathbb{Q}$ has a finite irrationality measure, in particular there are a finite number of $\frac{p}{q}\in\mathbb{Q}$ such that $$ \left|\pi - \frac{p}{q}\right|\leq \frac{1}{q^{10}}\quad\Longleftrightarrow\quad d(p,\pi\mathbb{Z})\leq\frac{1}{q^9} $$ and for any sufficiently large $n\in\m...

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Convergency of series The task is to prove, that series: $$\sum_{n=2}^\infty \frac{1}{log^2(n!)}$$ converges. Unfortunately I only managed to deal with showing that Limit of the summand is equal to 0, what is pretty o...

**Hint:** you can easily show that $\ln(n!) \geq c n\ln n$ for some absolute constant $c >0$. From there, $\frac{1}{\ln^2 n!} \leq \frac{1}{c^2 n^2\ln^2 n}$, and conclude by theorems of comparison.

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test the uniform convergency of the sequence of function Test the uniform convergency of the following sequence of functions in $[0,\pi]$. $$ f_n(x)=\frac{\sin nx}{1+nx}$$ Clearly we can see that in converges pointw...

If $x_n=\pi/2n$, then $f_n(x_n)=\frac{1}{(1+\pi/2)}$. Take $\epsilon = \frac{1}{2(1+\pi/2)}$.

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convergency of the sequence $x_n=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+(-1)^{n+1}\frac{1}{n}.$ > Test the convergency of the sequence $\\{x_n\\}$ , where $$x_n=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+(-1)^{n...

You may just use the Dirichlet test: $$ \frac1n \geq \frac1{n+1}, $$ $$ \frac1n \to 0, \, \text{as}\,\, n \to +\infty, $$ $$ \left|\sum_1^n (-1)^{k-1}\right| \leq 1 $$ ensuring the **convergence** of the series $$ \sum_1^\infty \frac{(-1)^{n-1}}{n}. $$

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Convergency of $\sum_{n=0}^{\infty}\frac{n!}{(kn)!}$, where $k > 1$ I am confident that $$\sum_{n=1}^{\infty}\frac{n!}{(2n)!}\approx1.5923$$ converges. Other series such as $$\sum_{n=1}^{\infty}\frac{n!}{(1.1n)!}\appr...

$\log\Gamma(s+1)$ is a convex function on $\mathbb{R}^+$ and $$ \frac{n!}{(kn)!} = \frac{\Gamma(n+1)}{\Gamma(kn+1)}\leq \frac{1}{n^{(k-1)n}}.$$ Something similar but more accurate can be deduced from Stirling's inequality $$ \left(\frac{m}{e}\right)^m\sqrt{2\pi m}\, e^{\frac{1}{12m+1}}\leq\Gamma(m+1...

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Proof convergency of series $a_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \ldots + \frac{1}{n!} $ I have used Cauchy and came to step where i have $\frac{1}{(n+1)!} + \frac{1}{(n+2)!} + \ldots + \frac{1}{(n+p)!} $ i cant f...

Hint: The "tail" is less than the sum of the geometric series $$\frac{1}{(n+1)!}\left(1+\frac{1}{n+2}+\frac{1}{(n+2)^2}+\frac{1}{(n+2)^3}+\cdots\right).$$

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Convergency of the power series at two points Consider the power series $$\sum_{n=0}^{\infty}a_{n}(z+3-i)^{n}.$$ The series converges at $5i$ & diverges at $-3i$. Then which is correct ? (a) convergent at $-2+5i$ ...

The series is a power series centered at $i-3$. It has a radius of convergence, unique by the theory. Inside that, the series converges absolutely, outside it does not. From the data you have (observe that $5i$ and $-3i$ have the same distance from the center $i-3$) you can infer that the radius of ...

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Testing the convergency of a sequence. > Question. Let $~f\in C^1[-\pi,\pi]$ be such that $f(-\pi)=f(\pi)$. Define $$a_n=\int_{-\pi}^{\pi}f(t)\cos nt~dt~, ~n\in \mathbb{N}$$ Which of the following statements are t...

For b) and c) you just have to integrate by parts. $a_n=-\frac 1 n \int_{-\pi} ^{\pi} f'(t) \sin (nt) \, dt$ so $na_n$ are the coefficients of the sine series of $g=-f'$ which is continuous and periodic. By standard results in the theory of Fourier series b) and c) are both true. In fact $\pi \sum n...

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Test the convergence of $\sum 1/(\log n)^{\log n}.$ I can not understand that how it is proved, so please somebody help me.I approch by logarithimic test but I unable to find out the convergency or divergency.

The Cauchy condensation test: $$\sum_{n=2}^\infty\frac1{(\log(n))^{\log(n)}}<\sum_{n=1}^\infty\frac{2^n}{(\log(2^n))^{\log(2^n)}}=\sum_{n=1}^\infty\frac{2^n}{n^{n\log(2)}(\log(2))^{n\log(2)}}<\sum_{n=1}^\infty\frac{2^n}{n^n}$$ And that last sum converges by ratio test, hence your series converges.

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How to test the convergency of a series How to test that the following series is convergent $$\frac{1}{2}+\frac{1}{2+1}+\frac{1}{2^2+1}+\frac{1}{2^3+1}+\dots$$ attempt: $$\lim_{n\to ∞}\frac{u_{n+1}}{u_n}=\lim_{n\to...

$$\lim_{n\to ∞}\frac{u_{n+1}}{u_n}=\lim_{n\to ∞}\frac{2^n+1}{2^{n+1}+1}$$ divide the Numerator and denominator by $2^{n+1}$ $$\lim_{n\to ∞}\frac{\frac{2^n}{2^{n+1}}+\frac{1}{2^{n+1}}}{\frac{2^{n+1}}{2^{n+1}}+\frac{1}{2^{n+1}}}=\lim_{n\to ∞}\frac{\frac{2^n}{2^{n}.2}+\frac{1}{2^{n+1}}}{\frac{2^{n+1}}{...

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Conceptual doubts on convergency of $a^x$. I am studying Convergency and Divergency of series in Higher Algebra by Hall and Knight. This article is particularly giving me a hard time. > _To show that the expansion of...

Let's break the question down to smaller parts. > expansion of $a^x$ in ascending powers of $x$ presumably means (Taylor expansion): $$a^x = 1 + (\ln a)x + \frac{\ln^2 a}{2}x^2+\dots = \sum_{n=0}^\infty \frac{((\ln a) x)^n}{n!}$$ Now, presumably > $u_n \triangleq \frac{((\ln a) x)^n}{n!}=\frac{((\ln...

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Test the uniform convergency of a series of function Consider the series of function $$\sum_{n=1}^{\infty}\frac{x}{1+n^2x}.$$ Show that this series of function is NOT uniformly convergent in $[0,1]$. I know only tw...

You may observe that, with $\displaystyle f_n(x)=\frac{x}{1+n^2x}$, $x \in [0,1]$, we have $$ (f_n(x))'=\frac{1}{(1+n^2x)^2}>0 $$ thus $\displaystyle f_n(x)$ is increasing with $x$: $$ 0<\sup_{[0,1]}f_n(x)=f_n(1)=\frac{1}{1+n^2} \leq \frac{1}{n^2}, $$since $\displaystyle \sum\frac{1}{n^2}$ is conver...

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Evaluate ( if convergent) $\int\limits_0^\infty \frac{1}{\sqrt{x(1-x)}} dx $ I would like to check whether the improper integral $$\int\limits_0^\infty \frac{1}{\sqrt{x(1-x)}} dx $$ is convergent or not. How can I che...

The function $1/\sqrt{x(1-x)}$ is only defined over $(0,1)$, so the proposed integral makes no sense. If you want to compute $$ \int_0^1\frac{1}{\sqrt{x(1-x)}}\,dx $$ your substitution is correct: set $x=\sin^2\theta$, with $\theta\in(0,\pi/2)$, so $$ dx=2\sin\theta\cos\theta\,d\theta $$ and you hav...

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Exponential(1) distributed random variable convergence I am stuck with convergency in probability... I have the following exercise: Let $(X_k)_{k\ge1}$ be a sequence of independent exponential-(1) distributed random ...

Let $X\sim Exp(1)$. So for $x\geq0$, $\mathbb{P}(X> x)=e^{-x}$. Let $X_i$ be iid to $X$ for $i=1,2,...$ and let $Y_n=n^\alpha \min\\{X_1,...,X_n\\}$ for some $\alpha x)=\mathbb{P}(\min\\{X_1,...,X_n\\}> xn^{-\alpha})=\mathbb{P}(X_1> xn^{-\alpha},...,X_n> xn^{-\alpha})$$ As the $X_i$ are identically ...

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convergency

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convergency

Convergency of $\sum_{n=1}^{\infty}\frac{\csc(n)}{n!}$ I am stuck on how to prove the convergency of the series $$\sum_{n=1}^{\infty}\frac{\csc(n)}{n!}.$$ It seems like that the series converges to approximately $2.85...

Convergency of series The task is to prove, that series: $$\sum_{n=2}^\infty \frac{1}{log^2(n!)}$$ converges. Unfortunately I only managed to deal with showing that Limit of the summand is equal to 0, what is pretty o...

test the uniform convergency of the sequence of function Test the uniform convergency of the following sequence of functions in $[0,\pi]$. $$ f_n(x)=\frac{\sin nx}{1+nx}$$ Clearly we can see that in converges pointw...

convergency of the sequence $x_n=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+(-1)^{n+1}\frac{1}{n}.$ > Test the convergency of the sequence $\\{x_n\\}$ , where $$x_n=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+(-1)^{n...

Convergency of $\sum_{n=0}^{\infty}\frac{n!}{(kn)!}$, where $k > 1$ I am confident that $$\sum_{n=1}^{\infty}\frac{n!}{(2n)!}\approx1.5923$$ converges. Other series such as $$\sum_{n=1}^{\infty}\frac{n!}{(1.1n)!}\appr...

Proof convergency of series $a_n = 1 + \frac{1}{1!} + \frac{1}{2!} + \ldots + \frac{1}{n!} $ I have used Cauchy and came to step where i have $\frac{1}{(n+1)!} + \frac{1}{(n+2)!} + \ldots + \frac{1}{(n+p)!} $ i cant f...

Convergency of the power series at two points Consider the power series $$\sum_{n=0}^{\infty}a_{n}(z+3-i)^{n}.$$ The series converges at $5i$ & diverges at $-3i$. Then which is correct ? (a) convergent at $-2+5i$ ...

Testing the convergency of a sequence. > Question. Let $~f\in C^1[-\pi,\pi]$ be such that $f(-\pi)=f(\pi)$. Define $$a_n=\int_{-\pi}^{\pi}f(t)\cos nt~dt~, ~n\in \mathbb{N}$$ Which of the following statements are t...

Test the convergence of $\sum 1/(\log n)^{\log n}.$ I can not understand that how it is proved, so please somebody help me.I approch by logarithimic test but I unable to find out the convergency or divergency.

How to test the convergency of a series How to test that the following series is convergent $$\frac{1}{2}+\frac{1}{2+1}+\frac{1}{2^2+1}+\frac{1}{2^3+1}+\dots$$ attempt: $$\lim_{n\to ∞}\frac{u_{n+1}}{u_n}=\lim_{n\to...

Conceptual doubts on convergency of $a^x$. I am studying Convergency and Divergency of series in Higher Algebra by Hall and Knight. This article is particularly giving me a hard time. > _To show that the expansion of...

Test the uniform convergency of a series of function Consider the series of function $$\sum_{n=1}^{\infty}\frac{x}{1+n^2x}.$$ Show that this series of function is NOT uniformly convergent in $[0,1]$. I know only tw...

Evaluate ( if convergent) $\int\limits_0^\infty \frac{1}{\sqrt{x(1-x)}} dx $ I would like to check whether the improper integral $$\int\limits_0^\infty \frac{1}{\sqrt{x(1-x)}} dx $$ is convergent or not. How can I che...

Exponential(1) distributed random variable convergence I am stuck with convergency in probability... I have the following exercise: Let $(X_k)_{k\ge1}$ be a sequence of independent exponential-(1) distributed random ...