## Transform the time series into base 100 in R

Is there a function that converts an entire time series into base 100 in R? If not, what would be the solution to the problem?
thank you so much

## real analysis – Acceleration of convergence of a series using optimal parameters

One of the ways to speed up the convergence of a series is to turn it into a faster series using optimal settings. Examples of this approach can be found in this article. I've got a generalization of this method to express a series in terms of two other independent variables:

Yes $$| frac {x} {x + y} | <1$$. Then,

$$sum_ {n = 1} ^ { infty} a_nx ^ n = Big ( frac {y} {x + y} Big) ^ {r + 1} sum_ {n = 0} ^ { infty} Big ( frac {x} {x + y} Big) ^ {n} sum_ {k = 0} ^ {n} {n + r choose k + r} a_k y ^ k$$

This expresses a series of powers in the LHS in terms of two independent variables $$y$$ and $$r$$ which, in theory, can be adjusted so that the series converges more quickly.

Question: How to choose the optimum $$y$$ and $$r$$ so that the RHS converges at its fastest pace?

Note: Asked this in MSE but received no answer in a week. Post it here and delete it from MSE to avoid duplicates.

## sequences and series – The limit of the ends of the interval according to \$ n \$

For $$n in mathbb {N_0}$$ Consider the following sequence of intervals:

begin {align} A_k &: =[k, k + 1), quad k in mathbb {N_0} \ frac {A_k} {2} &: = left[ frac {k} {2}, frac {k + 1} {2} right), quad k in mathbb {N_0} \ . \ . \ . \ frac {A_k} {2 ^ n} &: = left[ frac {k} {2 ^ n}, frac {k + 1} {2 ^ n} right), quad k in mathbb {N_0} end {align}

Clearly, for each $$n in mathbb {N_0}$$intervals $$frac {A_k} {2 ^ n}$$, $$k in mathbb {N_0}$$ are disjoint two by two and their union covers the interval $$[0infty)[0infty)[0infty)[0infty)$$. Repair some $$t in[0infty)[0infty)[0infty)[0infty)$$. For each $$n in mathbb {N_0}$$ there is a unique $$k_n in mathbb {N_0}$$ such as $$t in frac {A_ {k_n}} {2 ^ n}$$, that is to say.
$$frac {k_n} {2 ^ n} leq t < frac {k_n + 1} {2 ^ n}.$$
I would like to show that, for example, $$frac {k_n + 1} {2 ^ n} downarrow t$$ as $$n rightarrow infty$$. This seems intuitively clear because with the growth $$n$$ the ends of the interval are only getting closer. But how can one show this rigorously? It seems like showing that $$frac {k_n + 1} {2 ^ n}$$ is a decreasing sequence, its infimum being equal to $$t$$ should be enough.

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## night – Can I desaturate the incoming light with the help of a filter or a series of filters mounted on the lens?

I have a very difficult shot ahead. I photograph in the moonlight on a field meadow. I do not want to do post-work. I have to do everything in real time (for reasons that are too long, there are more things in the shot that will only complicate the question.)

My only option is to use filters when shooting, to capture our perception of the moonlight. The camera captures the moonlight as white light. Our eyes, however, perceive it as very desaturated with a slight hint of blue. I have a blue filter, but the cliché is too blue. I need to desaturate the light coming into the camera (almost black and white.) Is it possible to use lens filters to desaturate the light?

I can not film in black and white, because I'm not only exposing the moonlight, but also a girl with a glowing balloon in the scene. All this must be done in one exhibition.

Is it possible to desaturate the light using a filter or a series of filters?

Thank you guys.

More information: Thank you very much to all of you, you are right, but here's why and what I need. I want the girl and the balloons to be illuminated and the balloons will be different colors and shiny. So, she'll be in the color of glory, but she'll be sitting in the moonlight. To get the moonlight look, I can desaturate the camera settings and use a filter. But when it's time to remove the cap from the lens (shutter still open after capturing the moonlight) and expose the girl and the balloons, the camera will capture this one and the balloons in these same desaturated parameters. My camera will not change settings in middle pose, and although I think it can combine two shows, it's a more complicated way to get where I'm going. And post is almost impossible. To capture the color in the image, then darken and blue and desaturate the rest of the landscape is extremely flawed.

## sequences and series – Polynomials in the Pancake problem

I noticed something interesting in this chart. Columns can be expressed by polynomials of order k. I can not check if it's still a polynomial for $$k = 7$$.
$$k = 0: 1$$
$$k = 1: n-1$$
$$k = 2: n ^ 2-3n + 2$$
$$k = 3: n ^ 3-5n ^ 2 + 8n-5$$
$$k = 4: n ^ 4- frac {15} 2n ^ 3 + frac {29} 2n ^ 2 + 3n-17$$
$$k = 5: n ^ 5- frac {65} 6n ^ 4 + frac {173} 6n ^ 3 + frac {148} 3n ^ 2- frac {862} 3n + 265$$
$$k = 6: n ^ 6- frac {883} {60} n ^ 5 + frac {157} 3n ^ 4 + frac {2155} {12} n ^ 3- frac {4570} 3n ^ 2+ frac {42767} {15} n-967$$
$$k = 7: frac {1679} {1680} n ^ 7- frac {2765} {144} n ^ 6 + frac {21541} {240} n ^ 5 + frac {69163} {144} n ^ 4- frac {717821} {120} n ^ 3 + frac {1462277} {72} n ^ 2- frac {10388033} {420} n + 5037$$

If I adjust the $$n$$ to one where the first positive value occurs, it is:
$$k = 0: 1$$
$$k = 1: n$$
$$k = 2: n ^ 2 + n$$
$$k = 3: n ^ 3 + n ^ 2-1$$
$$k = 4: n ^ 4 + frac92n ^ 3 + n ^ 2- frac92n + 1$$
$$k = 5: n ^ 5 + frac {55} 6n ^ 4 + frac {31} 2n ^ 3- frac {14} 3n ^ 2-2n + 1$$
$$k = 6: n ^ 6 + frac {917} {60} n ^ 5 + frac {713} {12} n ^ 4 + frac {565} {12} n ^ 3- frac {5 } {12} n ^ 2 + frac {409} {30} n-3$$
$$k = 7: frac {1679} {1680} n ^ 7 + frac {142} {9} n ^ 6 + frac {192} {5} n ^ 5- frac {3743} {36} n ^ 4- frac {18917} {240} n ^ 3 + frac {14119} {36} n ^ 2- frac {50761} {140} n + 100$$

Questions:

Can we prove that the columns of this table are always polynomials?

Perhaps the coefficients predicted in one way? There seems to be an order but I can not find it.

## Real Analysis – The Show Power series converges for every \$ x \$.

Let $$f (x) = 1 + a_ {1} x ^ {1} + a_ {2} x ^ {2} + a_ {3} x ^ {3} + a_ {4} x ^ {4} +. ..$$ to be a solution of the differential equation $$f (x) = xf (x).$$
Now, I have to explain that the power series that defines $$f (x)$$ converges for each $$x$$.

This problem is introduced by two smaller proofs that I am going to give (or my attempts to get them) for the thought trail to be clear.

I first found a simple expression for $$frac {a_ {2n}} {a {2n-2}}$$ for each $$n in mathbb {N}$$. Since this fraction is $$frac {1} {2}$$ when $$n = 1$$, he is $$frac {1} {4}$$ when $$n = 2$$, he is $$frac {1} {6}$$ when $$n = 3$$ and so on, I leave

$$frac {a_ {2n}} {a_ {2n-2}} = frac {1} {2n}$$

Then I had to show that for each fixed $$x in mathbb {R}$$ there is a $$N in mathbb {B}$$ such as $$| a_ {2n} x ^ {2n} | leq frac {1} {2} | a_ {2n-2} x ^ {2n-2} |$$ does not matter when $$n geq N$$. This is my attempt:

Let $$x in mathbb {R}$$ to be given and leave $$N> x ^ {2}$$then for all $$n geq N: frac {1} {n} < frac {1} {x ^ {2}}.$$ then

$$frac {| a_ {2n} |} {| a_ {2n-2} |} = frac {1} {2} frac {1} {| n |} < frac {1} {2} frac {1} {| x ^ {2} |} = frac {1} {2} left | frac {x ^ {2n-2}} {x ^ {2n}} right | \ Longrightarrow \ | a_ {2n} || x ^ {2n} | < frac {1} {2} | x ^ {2n-2} || a_ {2n-2} |.$$

I'm waiting to have to use $$| a_ {2n} x ^ {2n} | leq frac {1} {2} | a_ {2n-2} x ^ {2n-2} |$$ to show convergence, but I do not know how. No suggestion?

## deep learning – predicting time series with variable input length

My thesis focuses on the prediction of cancer in mice. I collected data from 35 mice. I measure the volume of tumors every day after the onset of cancer until the death of the mouse. The time of death varies between 50 and 72 days and so I have a series of 35 episodes of different length.

I have to predict the evolution of tumor volume over time. I want to use regression but I do not know how to adapt a model to 35 time series of different lengths.

note that I can not change the length of my dataset because I am losing important information about cancer behavior.

a suggestion for my problem?

## memory – which graphics cards can I upgrade to if I have an HP Slimline Desktop PC 270-p0xx series computer?

I am looking to upgrade the graphics card on my computer because I am currently running on Integrated Graphics. It would be nice to know what I could upgrade and find out if I need a more powerful fan and power supply. Here are my current specifications:

Processor: Intel Core i7-7700T

RAM: 12 GB

Graphics: Intel HD Graphics 630

I do not know my current power supply, so any help would be welcome. Thank you!

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