All Questions
Tagged with summation power-series
362
questions
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Infinite power summation
I was playing around with fractions in my head when I realized the infinite summation of $\frac{1}{b^n}=\frac{1}{(b-1)}$ for when $b>1$.
I tried writing a proof for this and researching the ...
user460386
1
vote
2
answers
2k
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Power series and shifting index
First I have to find the power series represantion for the following function:
$$\ f(x) = \ln(1+x)$$
I tried the following:
$$\ \frac{d}{dx}\Big(\ln(1+x)\Big) = \frac{1}{1+x} = \sum_{n=0}^\infty(-...
- 215
2
votes
1
answer
480
views
How to derive the Asymptotic series of sums related to Euler's Summatory Function?
I need techniques to solve the density of $T$, a subset of $\mathbb{Q}$ in form of an albegraic expression with relatively prime numerator and denominator values. The best way of doing this is by ...
- 1
4
votes
0
answers
828
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Laurent expansion for $\sqrt{z(z-1)}$
Let $f(z) = \sqrt{z(z-1)}$. The branch cut is the real interval $[0,1]$, and $f(z)>0$ for real $z$ that are greater than 1. I need to find the first few terms of the Laurent expansion of $f(z)$ for ...
- 1,303
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votes
3
answers
63
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Sum $\sum_{i=1}^n\frac{3^{i-1}}{2^{i-2}}$
$$\sum_{i=1}^n\frac{3^{i-1}}{2^{i-2}}$$
Currently, I am just iterating i from 1 to n but ...
- 369
1
vote
1
answer
1k
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Integral of Summation (power series)
Could someone guide me through this process, I am confused on how you can take an integral of the factorial or whatever is going on in the problem.
In the context of this problem, the summation is ...
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1
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1
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66
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Values for which this sum can be defined in terms of known constants in a closed form
I'm interested in the sum,
$$\sum_{n=1}^\infty\frac{\zeta(2n)\Gamma(2n)}{\Gamma(2k+2n+2)}x^{2n}$$
Otherwise written as
$$\sum_{n=1}^\infty\frac{\zeta(2n)}{(2n)(2n+1)\cdots(2n+2k+1)}x^{2n}$$
I am ...
- 3,557
2
votes
1
answer
569
views
Find the sum of the series $\frac{1}{2^a 3^b 5^c}$
I'm trying to compute this sum:
$$\sum_{1 \le a \lt b \lt c; a,b,c \in\mathbb N}^n \frac{1}{2^a 3^b 5^c}$$
I've tried to try to compute $\sum\frac{1}{2^{a+b+c}}$ and $\sum \frac{1}{5^{a+b+c}}$ (...
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votes
2
answers
113
views
Calculate the sum of a power series
If $$f(x)=\sum_{k=0}^\infty a_k (x-1)^k$$
For $|x-1|<r$. Find the sum of the following series, and the values of x for which it converge
$$\sum_{k=3}^\infty \frac{(2k-1)a_k}{k^3-4k}(3x+2)^k$$
I ...
- 185
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votes
2
answers
102
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Find the sum of the following infinite series $e^{-x}\sum_{i=0}^{\infty}\frac{i.x^i}{i!}$
Find the sum of the following infinite series $$e^{-x}\sum_{i=0}^{\infty}\dfrac{i.x^i}{i!}$$
The summation looks like an exponential series but how to tackle that?$$ 0+\frac{x}{1!}+\frac{2x^2}{2!}+......
- 586
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votes
3
answers
74
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Infinite series $\sum_{n=2}^\infty n(\frac{1}{2})^{n-2}$ [closed]
I am a little confused about calculating $\sum_{n=2}^\infty n(\frac{1}{2})^{n-2}$. I know that the infinite series $\sum_{n=1}^\infty nz^{n}$ yields $\frac{z}{(1-z)^2}$. However, now I start at $2$ ...
- 341
1
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2
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To what value does this summation converge: $\sum_{r=o}^{n}{\frac{\binom{n-1}{r}r!}{n^{r+1}}}$
I had been trying to solve (Probability of rolling a 1 before you roll two 2's, three 3's, etc) this problem for quite some while and I think I had found some way ahead but I cant seem to find ...
- 2,758
4
votes
4
answers
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What is the sum of $\sum_{k=0}^\infty \frac{(-1)^{k+1}}{2k+1}$?
What is the sum of the following expression:
$$\sum_{k=0}^\infty \frac{(-1)^{k+1}}{2k+1}$$
I know it is convergent but I cannot evaluate its sum.
- 24.5k
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votes
1
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Calculate $\sum^{\infty}_{n=1}n^2x^{n-1}$ [duplicate]
I have some struggles with this exercise. I need to find out $$\sum_{n=1}^{\infty} n^2x^{n-1}$$ I know that the answer is $\frac{1+x}{(x-1)^3} $ when $|x|<1$ $ $. And I need to solve it by using ...
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2
votes
2
answers
219
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Summation of alternating series, Mercator series: $\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n(n+1)}$
I am struggling with solving sum of this alternate series:
$$
\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)}\
$$
I know that:
$$
\log(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \cdot x^n\
$$
But ...
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8
votes
1
answer
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Summation calculus: $\sum_{k=1}^n \frac{2^{2^{k-1}}}{1-2^{2^k}}$
How can I solve this?
$$\sum_{k=1}^n \frac{2^{2^{k-1}}}{1-2^{2^k}}$$
Actually I tried many direction, but failed.
Please give me some right direction.
$$\sum_{k=1}^n \frac{2^{2^{k-1}}}{1-2^{2^k}} = \...
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1
answer
160
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Upper bound of sum of sequences
The sum is $\sum_{i=1}^{\infty} \frac{a^n}{n}$, where $0<a<1$. One easy upper bound is $\frac{a}{1-a}$. Are there any tighter upper bound available?
- 1,001
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1
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How can you use combinations to find a general formula for the power series summation
I do not understand the fundamental theorem of calculus so I have been trying to find a proof that the antiderivative of a function evaluated at $x=b \text{ and } x=a$ gives the area under the curve ...
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0
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Difficult De Moivre's theorem question involving series
Use De Moivre's theorem to show that
$$\sin (2m+1) \theta = (\sin^{2m+1} \theta) \cdot P_m (\cot^2 \theta)$$ for $0 < \theta < \pi/2$, where
$$P_m(x) = \sum_{k=0}^{m} (-1)^k C^{2m+1}_{2k+1} ...
- 720
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votes
2
answers
118
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represent the following in a power series
Represent $1/(1-x)^2$ in a power series using the fact that $$\ln (x+1)= \sum_{n=1}^\infty (-1)^{n-1} \frac{x^n}{n}$$
First i derived $\ln(x+1)$ and its sum which equaled:
$$1/(x+1)= \sum_{n=2}^\...
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votes
2
answers
89
views
Let $T_n=\sum\limits_{i=1}^n\frac{(-1)^{i+1}}{2i-1}$ and $T=\lim T_n$, show that $\sum\limits_{n=1}^\infty(T_n-T)=\frac{\pi-2}8$ [closed]
Let $T_n=\sum\limits_{i=1}^{n}\frac{(-1)^{i+1}}{2i-1}$ and $T=\lim\limits_{n \to\infty}T_n$, show that $\sum\limits_{n=1}^\infty(T_n-T)=\frac{\pi}{8}-\frac{1}{4}$.
Attempt:
I know that $Im(\frac{1}{...
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5
votes
1
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Determine whether or not $\exp\left(\sum_{n=1}^{\infty}\frac{B(n)}{n(n+1)}\right)$ is a rational number
Let $B(n)$ be the number of ones in the base 2 expression for the positive integer n.
Determine whether or not $$\exp\left(\sum_{n=1}^{\infty}\frac{B(n)}{n(n+1)}\right)$$ is a rational number.
...
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3
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467
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Series with Binomial Coefficients
I need to get a closed form for this series $$\sum_{x=0}^{\infty} x {z \choose x} \lambda ^ x \mu^{z-x}$$
I know that that $\sum_{x=0}^{\infty} {z \choose x} \lambda ^ x \mu^{z-x} = (\lambda + \mu)^z$...
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3
votes
1
answer
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Power Series proofs
$$\alpha(x) =\sum_{j=0}^\infty \frac{x^{3j}}{(3j)!}$$
$$\beta(x) = \sum_{j=0}^\infty \frac{x^{3j+2}}{(3j+2)!}$$
$$\gamma(x) = \sum_{j=0}^\infty \frac{x^{3j+1}}{(3j+1)!}$$
Show that $\alpha(x+y) = \...
- 221
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votes
1
answer
224
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Closed form of the sum of the product of three binomial coefficients
I encountered with this kind of series from the calculation in quantum optics:
$$\sum_{n,m=0}^\infty \sum_{k,l=0}^{\min(n,m)}\binom{n}{k}\binom{m}{l}\binom{n+m-k-l}{m-k}A^{n+m}B^kC^l$$
Provided that ...
- 1,110
3
votes
2
answers
2k
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Power Series involving Double Factorials
This comes from my partial solution to another question. I need to find a closed form for the following summation
$$\sum_{n=1}^{\infty}\frac{n!}{(2n+1)!!}n^x$$
Where $x$ is a fixed integer
This ...
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3
answers
735
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Power of Series [duplicate]
In taking the power of a series
$$\left(\sum_{k=0}^{\infty} a_k x^k \right)^n =
\sum_{k=0}^{\infty} c_k x^k$$
do you know an expression for $c_k$ solely in terms of the coefficients $a_k$?
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1
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Sum of the unknown power series
I have the following series where $h$ and $f$ are some functions.
$$1+\frac{2}{5}fh^2+\frac{4}{21}f^2h^4+...,$$
which I figured can be written as
$$3\sum_{k=0} \frac{(2fh^2)^k}{4^{k+1}-1}.$$
I need ...
- 119
1
vote
1
answer
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Solving differential equation with infinite series sum
We are given the following differential equation,
$$x \cdot \frac{d^2y}{dx^2} + (2+x)\frac{dy}{dx}+y=0$$
i) Show the following,
$$x \cdot \frac{d^2y}{dx^2} + (2+x)\frac{dy}{dx}+y=\sum{[(n+1)c_n+(n+...
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2
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Sum of manipulated geometric series
Find the sum of $$\sum_{n=1}^{\infty} \frac{n^2}{2^n}$$
I know I need to manipulate the power series $\sum_{n=0}^{\infty}x^n$ with $x = \frac{1}{2}$, but I'm not sure how. Would differentiating it ...
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2
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161
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How to find coefficients in this sum?
$$\sum_{n=0}^{\infty} (-1)^{n}(\frac{x^{2n}}{(2n)!}) * \sum_{n=0}^{\infty} C_{n}x^{n} = \sum_{n=0}^{\infty} (-1)^{n}(\frac{x^{2n+1}}{(2n+1)!})$$
How to find coefficients until the $x^{7} $ ? It is ...
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1
answer
49
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Calculating the sum of a series (power series)
By expanding $(r+1)^4 -r^4$ calculate $\sum r^3$
step 1 - expand to get $1 + 4r + 6r^2 + 4r^3 $
step 2 - what should I do next to calculate the sum?
I don't really understand why we have been ...
- 25
6
votes
1
answer
280
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Sum of $\sum_{n \geq 1} \frac{(\ln x +1)^n}{n^n}$
I want to find the sum of the following series
$$\sum_{n \geq 1} \frac{(\ln x +1)^n}{n^n}$$
Using theorems on integration and differentiation of series. I can set $t=\ln x+1$ so that I get
$$\sum_{...
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1
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1
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How do I evaluate $\sum_{k=1}^nk^pr^k=?$
For this entire post, we have $r\ne1$, $n\in\mathbb N$. For the first half, $p\in\mathbb N$, and at the end $p\in\mathbb Q$.
It is well known that
$$\sum_{k=1}^nr^k=\frac{1-r^{n+1}}{1-r}$$
And
$$\...
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3
answers
74
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Calculating the sum $\sum_{n=1}^\infty \frac{n}{(n-1)!}x^n$?
So I've got the sum $$\sum_{n=1}^\infty \frac{n}{(n-1)!}x^n$$
To show that it converges for all real numbers, I used the ratio test. And found the convergence radius to be $$R = \frac{1}{L}, \qquad R ...
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0
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Proving an identity with geometric series.
I've been at this for MANY hours and I think it's time I sought help.
Question: Given $k = \frac{2 \pi}{Na}\left ( p-\frac{N}{2} \right )$, prove that $\sum_{k=1}^{N}e^{ika\left ( n-m \right )}=N\...
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3
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Formula for finite power series
Are there any formula for result of following power series?
$$0\leq q\leq 1$$
$$
\sum_{n=a}^b q^n
$$
- 303
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votes
11
answers
127k
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The idea behind the sum of powers of 2
I know that the sum of powers of $2$ is $2^{n+1}-1$, and I know the mathematical induction proof. But does anyone know how $2^{n+1}-1$ comes up in the first place.
For example, sum of n numbers is $\...
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1
answer
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Find the domain of convergence of the series $\sum^{\infty}_{n=1}\frac{n!x^{2n}}{n^n(1+x^{2n})}$
Find the domain of convergence of the series $\sum^{\infty}_{n=1}\frac{n!x^{2n}}{n^n(1+x^{2n})}$.
Using the ratio test, I got $ \left | \frac{x+x^{2n+1}}{1+x^{2n+1}} \right | $, but I don't know how ...
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1
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Studies about $\sum_{k=1}^{n} x^{\frac 1k}$
Are there any studies about this function?
$$f(x,n)=\sum_{k=1}^{n} x^{1/k}=x+x^{1/2}+x^{1/3}+x^{1/4}+\cdots +x^{1/n}$$
EDIT:
My first notes about it.
$f(1,n)=n$
$f'(1,n)=H_n$
$\int_0^1 \frac{f(x,...
- 611
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votes
3
answers
62
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Sum of Geometric infinite series [duplicate]
How do I solve this:
$$\sum_{k=1}^{\infty}k(1-p)^{k-1}$$
I forgot how to do this, or the formula I need to use. Could not find it online for some reason.
- 6,525
10
votes
2
answers
295
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Finding the sum $\frac{x}{x+1} + \frac{2x^2}{x^2+1} + \frac{4x^4}{x^4+1} + \cdots$
Suppose $|x| < 1$. Can you give any ideas on how to find the following sum?
$$
\frac{x}{x+1} + \frac{2x^2}{x^2+1} + \frac{4x^4}{x^4+1} + \frac{8x^8}{x^8+1} + \cdots
$$
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Show that $\ln\frac23=\sum\limits_{n=1}^\infty\frac{(-1)^n}{2^nn}$
I am working on the following problem,
I have managed to first prove that the series is convergent, using the conditions for the alternating series test, although am unsure how to find the exact sum. ...
2
votes
1
answer
428
views
Limit of a sum of powers [duplicate]
I need to find the limit of a sequence indexed by $n\in \Bbb N$. $k$ is fixed natural constant.
The sequence is:
$x_n = \frac{1^k+2^k+3^k+\ldots+n^k}{n^k} - \frac{n}{k+1}$
I tried to solve this ...
- 131
1
vote
2
answers
244
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Series summation of polynomial on x multiplied by x power summation
An unbiased coin is tossed repeatedly and outcomes are recorded. What is the expected no of toss to get HT ( one head and one tail consecutively) ?
I tried to solve above question using the following ...
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0
votes
2
answers
219
views
Evaluate $\sum_{k=1}^{\infty} (-1)^{k+1}\frac{\pi^{6k}}{6k!}$
Evaluate
$$\sum_{k=1}^{\infty} (-1)^{k+1}\frac{\pi^{6k}}{(6k)!}$$
I was trying to find a closed form for this sum
$$\sum_{k=1}^{\infty} (-1)^{k+1}\frac{x^{6k}}{(6k)!}$$
I believe there is ...
- 946
5
votes
3
answers
143
views
How can I get a good approximation of $\sum_{s=1}^\infty x^s\ln(s)\ $?
How can I get a good approximation of the sum
$$\sum_{s=1}^\infty x^s\ln(s)$$
by hand ?
If I only consider the part of the function, where it is strictly decreasing, I can bound the sum by ...
- 85.1k
1
vote
0
answers
201
views
Find the Laurent Series expansion.
Find the Laurent series expansion of $$f=\frac {2z}{(z-1)(z-3)}$$ at the $1<|z|<3$.
Sol.: My centre is around zero.Ill use the known geometric series around zero.
$$f(z)=\frac {2z}{(z-1)(z-3)}=...
- 2,782
5
votes
3
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963
views
Sum identity using Stirling numbers of the second kind
Experimenting with series representations of $e^{x e^x}$ I came across the two seemingly different power series
$$e^{x e^x} = \sum_{n=0}^{\infty} x^n \sum_{k=0}^{n} \frac{(n-k)^k}{(n-k)! \cdot k!}$$
...
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0
votes
1
answer
419
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Order of summation for shifted exponential function
I want to represent the function:
\begin{equation}
f(x)=e^{-a(x-b)^{2}}
\end{equation}
where, $0<a<1$, $x\in\mathbb{R}$, and $b\in\mathbb{R}$.
As a power series for an integral I am working ...
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