Wikipedia:Reference desk/Archives/Mathematics/2012 January 12

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January 12

Power series reloaded

Hey again, sorry for so many questions! I will help someone else when I can, I promise! My question is, where does the power series for an exponential function a^x arise naturally? Alternately put, assuming we know nothing about Taylor series but assuming that we know infinite series "work", how can we discover it without constructing it from its derivatives? (I think the series for this particular function was known before Taylor series were developed, so someone apparently did it) What I'm thinking is something along the lines of a formula or discovery or something where it just "appears" (and possibly we do not know it is the series for a^x). Thanks, and sorry if I wasn't really clear. 24.92.85.35 (talk) 02:53, 12 January 2012 (UTC)

The exponential function y=e^x satisfis the linear differential equation y'=y and the initial condition y(0)=1. Substituting the power series y=Σ a_ixⁱ into the initial condition gives a₀=1. Substituting the power series into the differential equation gives Σia_ixⁱ⁻¹=Σa_ixⁱ=Σa_i−1xⁱ⁻¹. So ia_i=a_i−1. So a_i=a_i−1/i=a₀/i!=1/i!. So e^x=Σxⁱ/i! Bo Jacoby (talk) 08:22, 12 January 2012 (UTC).

Looks like the OP was asking about $a^{x}$ for a general a. But this is kind of moot since to expand this you usually first translate it to $\exp(x\ln a)$ and use the power series for exp. And the series for that can be found for example using exp'=exp or $e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}$. Both ways are fairly elegant, I don't think you'll find something more "natural". -- Meni Rosenfeld (talk) 10:38, 12 January 2012 (UTC)

A variation of Bo Jacoby's answer: the differential equation y'=y with IC y(0)=1 is equivalent to y=1+Ty where $\scriptstyle Ty(x)=\int _{0}^{x}y(t)dt$. We may write it as $\scriptstyle (I-T)y=1$ and solve in y using the Neumann series expansion $\scriptstyle (I-T)^{-1}=\sum _{k=0}^{\infty }T^{k}$, so $\scriptstyle y(x)=\sum _{k=0}^{\infty }T^{k}[1]$. Here $\scriptstyle T^{k}[1]$ is the k-th iterated integral of the constant 1, that is $\scriptstyle x^{k}/k!$, and we get the exponential series. So in a certain sense the exponential series is an instance of the geometric series. --pma 08:39, 14 January 2012 (UTC)

How advantageous is it to replace a paper-towel dispenser with a wall dryer?

In a dorm restroom, how often does an average stack of paper towels get used up, how much is it to get another new stack, and therefore, how much will it cost per month and per year to keep the typical paper towel dispenser restocked?

On the other hand, if it gets replaced with a heated wall dryer, how much electricity would the typical new 2012-model wall dryer consume in each typical use, in a month, and a year? (Let us assume 9¢ a kilowatt-hour, as it was 8.44¢/KWh in Kansas last time I checked in May.)

(The initial purchase of a dryer will be made by donation, so the recipient won't have to worry about the procurement cost.)

Therefore, how far ahead will the dorm come out by switching from a paper towel dispenser to a heated wall-dryer? Thanks! --70.179.174.101 (talk) 13:23, 12 January 2012 (UTC)

How many paper towels are in an average stack? It depends. How often is an average stack used up? It depends. How often is the stack refilled? It depends. How much does a new stack cost? It depends. How much does the electricity for a heated dryer cost? It depends. That is the only thing that you gave values to work with. You need to make assumptions about everything else. Otherwise, any result is simply useless. -- kainaw™ 13:51, 12 January 2012 (UTC)

You could get an energy monitor to measure actual energy usage, or estimate it using the power rating of the device and multiply it by the time the drier is on. Finding out how much energy is used to make a paper towel is a harder, you could start at Carbon footprint to try and find out.--Salix (talk): 14:22, 12 January 2012 (UTC)

Another option I thought of - if this is real life - is to simply install the air dryer next to the paper towel dispenser. If the air dryer saves money and some people actually use it, then there will be an overall savings. Of course, there would be more savings if everyone simply stopped washing their hands! -- kainaw™ 14:27, 12 January 2012 (UTC)

Shouldn't questions like this be under Miscellaneous rather than maths? The work here is looking up things and knowing about the objects, why would anyone on this reference desk know anything like that? Dmcq (talk) 14:35, 12 January 2012 (UTC)

Well I did a bit of looking about and its tricky to find comprehensible data. The European Commission give full lifecycle inputs and output for some materials such as Corrugated board boxes, slightly more useful is the UK Enviornment Agency who have spreadsheet of CO2 equivilents for the construction industry[1]. Nearest equivalent to a paper towel I could find on that is particle board at 0.54 tonnes of CO2 per tonne of material. Grid electric is 0.00059368 tonnes of CO2 per kWh or 593.68g CO2 per kWh. I estimate 1 paper towel is 5g so 2.7g of CO2 equivalent (excluding transport costs). I estimate air drier runs at about 3kW (same as my kettle) and I spend 30 secs drying my hands using it, so thats 0.025 kWh or 15g of CO2. My estimates could easily be off by an order of magnitude so I can draw any conclusion.--Salix (talk): 15:48, 12 January 2012 (UTC)

The OP didn't raise any question about carbon, but about cost of the $ & £ sort. --Tagishsimon (talk) 15:54, 12 January 2012 (UTC)

Another issue is that paper towels can do things which hot air blowers can't, like allow you to open the bathroom door without having germs redeposited on your hand, or clean up water splashed all over the sink. So, if you remove the paper towels, you may find that toilet paper is now used for those purposes, and it's usage increase must then enter into the calcs. So, I second the idea of offering both the blower and towels. Paper towels can also both be recycled and made from recycled paper, so that's a plus. StuRat (talk) 16:42, 12 January 2012 (UTC)

My reply-all

Okay, I cannot calculate the paper towel usage myself, so try a typical 54-person dorm. There's a girl's floor, a guy's floor, and a "grab-bag" floor, each with a restroom. Can we establish an assumption with said dorm? If anyone has lived in or worked with a dorm, then they may be able to give a reasonable ballpark. I'm not looking for "carbon footprint" because the costs matter more to them, but I would have to assume the footprint is less with a wall dryer. (The outlet could turn in many ways and therefore even help dry a nearby sink, for example.)

About the cost of bundles of paper towels, let's choose a supplier easily available in the state of Kansas, preferably to the town of Manhattan. We can figure in their cost.

A dryer takes 30 seconds to dry, typically. Some decide to take a full minute. How many kWh are used in one minute? (Remember the figure of 9¢/kWh was cited.) Was anything else needed or are we good?

(Moreover, I hope my reply was well-formed enough because after having done plenty today, I am about ready to nap, so I'm not sure how my mental processes are right now. But I hope this was a good enough shot.) --70.179.174.101 (talk) 00:49, 13 January 2012 (UTC)

You need to estimate how often each person uses the restroom. This will depend on the nature of the dorm and the people within: if they're out all day and not back till late in the evening (or sleeping over with their significant others in deluxe off-campus accommodation, or visiting their parents for days at a time) then they may only use it twice a day, but if they're in the dorm all day studying hard and drinking lots of coffee then they may use it 8 or 10 times a day each. (This may of course change at different times of year.) You are far better placed than anybody else on this website to measure the usage of paper towels in your dorm. --Colapeninsula (talk) 11:17, 13 January 2012 (UTC)

You could calculate the difference in cost per visit. Assuming a 3kW drier running for 30 sec, that 0.025 kWh. at 9¢ a kilowatt-hour so 0.225¢ per use. In the UK 15 packs of 200 towels cost £27.60 about $42 which works out at about 1.4¢ per towel. Its looking like a 1¢ saving per use, more if people use two towels, less if they spend longer using the direr. Just multiply this by the estimated usage. You could probably workout the actual usage just by seeing how many packs of towels are used each week.--Salix (talk): 16:05, 13 January 2012 (UTC)