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Questions tagged [convergence-divergence]

Convergence and divergence of sequences and series and different modes of convergence and divergence. Also for convergence of improper integrals.

This tag is for questions about Convergent Sequences and Series and their existence, and by extension also for divergent ones. Also for convergence/divergence of improper integrals.

Formally, a sequence $S_n$ converges to the limit $S.$

$$\lim_{n\to \infty}S_n=S $$ if, for any $\epsilon>0$, there exists an $N>0$ such that $|S_n-S|<\epsilon$ for $n>N$.

A sequence diverges if it does not converge.

For more specialized questions about divergent series, such as summation methods, consider the tag .

21990 questions
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Is this demonstration of convergence not a bit too complicated?

Let be $$g_n(x)=\begin{cases} ~0 & \text{if } x \leq 0 \\ ~1 & \text{if } x=\frac{1}{n}\\ ~0 & \text{else} \end{cases}%$$ $g$ is affine and continuous if $x\leq 0$: $g_n(x)$ is constant and equals to $0$. if $x\geq 0$ then let be $N$ such that…
Revolucion for Monica
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Prove that if a real convergent sequence with limXn=x as n--> infinity, if $X_n<6$, then $x\le6$ and if $X_n>2$ then $x\ge2$

I am very new to this and I am finding it very difficult to do proofs, here is my attempt. If for each $n\in\mathbb N$ $X_n<6$, prove that $x≤6$ Let $\varepsilon > 0$ be given. since $(Xn)$ converges $|X_n-x|<\varepsilon$ and we are told $X_n<6$…
user299113
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Convergence of a Series Of Functions

We've been starting to talk about series approximations in my numerical analysis course, and I got to this one series question that has been stumping me. For $n \in \mathbb{N}$ (assume $\mathbb{N}$ doesn't contain a $0$) and $x \in \mathbb{R}$…
Josh
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How to show convergence of a particular series

Knowing that $\sum_{n=1}^\infty 3^{-n}$ converges, how does one show that $\sum_{n=1}^\infty 3^{-n} \cdot n$ converges? I know the first series converges by the comparison test (comparing with $\sum_{n=1}^\infty 2^{-n}$), but for the second series…
Alhashemi
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Convergence of numerical method without stability?

It is straight-forward to prove convergence of a numerical method given consistency and stability, but when does this break down? The proof for convergence says: \begin{equation} |y^{n} - \hat{y}^{n}| \leq \left(\sum_{p=0}^{n-1} \sigma ^{p} \right)…
mfordcc7
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Proving that a product converges

I have to prove that the following product converges: $\prod\limits_{k=1}^n \frac{2k-1}{2k}$ I've seen convergence of sums before, but this is new to me. In fact, I had to look up the $\Pi$ notation. I figure I can't use the usual approaches here…
John
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Proving convergence of sum to 2n

I need to prove that the following sequence converges as $n\to\infty$: $\sum\limits_{k=n+1}^{2n} 1/k$ The problem is that I've only ever seen sums from i to n for example. I'm confused because not only is 2n alien to me, but also k and n are…
John
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Convergence rate for $a_{n+1}=\sqrt{2 \sqrt {a_n}}$?

Convergence rate for $a_{n+1}=\sqrt{2 \sqrt {a_n}}$ I don't know the next step after :$$\frac {a_{n+1}^4}{ a_n}={4}$$ Edit, maybe there is no simple answer for this, is there a rate of convergence known for $a_{n+1}=\sqrt{2 {a_n}}$ aka $$\sqrt {2…
jimjim
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L2- Convergence vs. pointwise convergence

I am reading about convergence of Fourier Series and the author uses a equality symbol "$=$" in the following equation instead of the usual "$\rightarrow$" to represent the convergence behavior of a error function $e(t)$. My question is: In the…
User1983
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Show that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$

Could someone please show me the algebraic steps in showing that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$? As the way I see it $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \infty$ as $x \rightarrow \infty$ Thanks in…
BLAZE
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Alternating series convergence test

I have a series $\sum_{n=1}^\infty c_n x^n$ where $c \le c_n \le C$. I can determine radius of convergence easily by the root test, but how does one determine convergence for $x = -1$? It is not a positive series, so divergence test does not work,…
Peter Felafe
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uniform converging functions

If I know that $\sum_M^\infty v(x)$ converges towards $f$, then $$\left| \left( f + \sum_{1}^{M-1} v(x) \right) - \sum_{1}^N v(x)\, \right| = \left|\;f - \sum_{M}^N v(x)\,\right| < \epsilon$$ for large enough $N$, and so our general series converges…
Shesai Elias
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Series: only the latter terms matter

I've been told that when it comes to uniform convergence of series, only the tail matters, This seems intuitively obvious, but is there a theorem one can refer to? Further, if $\sum_{m}^\infty f_n(x)$ converges uniformly towards $f$, does $\sum_{n =…
Cleese Vanderborckenhug
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for which alpha is the Integral convergence

Let $\alpha>0$ and $$ f(x)=\frac{\ln x}{(x-1)^{\alpha}} $$ for $x>1$ i found that for $\int_2^{\infty}f(x) dx$ the integral is convergence for $\alpha > 2$ but for which $\alpha$ is $f(x)$ convergence for $\int_1^2f(x) dx$ and $\int_1^{\infty}f(x)…
Jamgreen
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How to show that Newton's method has the linear convergence rate with 1-1/m?

How to show that Newton's method has the linear convergence rate with 1-1/m ? (For a zero of multiplicity m>=2)
Tsing
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