Wikipedia:Reference desk/Archives/Mathematics/2016 October 7

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October 7

What is the value of these sums

$\sum _{k=0}^{\infty }{\frac {1}{k\times 10^{100}+1}}$

and

$\sum _{k=0}^{\infty }{\frac {1}{k^{k+1}\times 10^{100}+1}}$

175.45.116.99 (talk) 02:41, 7 October 2016 (UTC)

The first sum diverges, and the second is extremely close to 1 ($\lim _{x\rightarrow \infty }\sum _{k=0}^{\infty }{\frac {1}{k^{k+1}\times x+1}}=1$). 24.255.17.182 (talk) 04:06, 7 October 2016 (UTC)

To elaborate just a bit, the first one is $\sum _{k=0}^{\infty }{\frac {1}{k\times 10^{100}+1}}\approx {\frac {1}{0+1}}+\sum _{k=1}^{\infty }{\frac {1}{k\times 10^{100}}}=1+{\frac {1}{10^{100}}}\sum _{k=1}^{\infty }{\frac {1}{k}},$ where the last sum is the divergent harmonic series. Loraof (talk) 16:20, 7 October 2016 (UTC)

The value of the second is approximately $1.000000000....$ (99 zeroes in total) $....000013839$ Dbfirs 00:03, 11 October 2016 (UTC)

Beats me

I’m not crash hot with math, and confess this claim by an oil company is doing my head in. Their promotion/exercise is based on a price of $2 per litre, and a discount of 6 cents per litre. Basically, they claim that by accumulating discounts rather than redeeming them at the time of sale, you will double your savings. Following are their examples given.

Redeemer. Buys 50 litres of gas, value $100, but pays $97 because of the 50 x 6c discount. After doing this four times the redeemer has spent $400 for a total savings of $12.

Accumulator. This is complicated because the promoter hasn’t exactly compared apples with apples. What they show is (a) someone making seven $40 purchases but accumulating the 6c litre discounts each time, and (b) then buying $100worth of gas for only $76 which they say incorporates all of the accumulated discounts and the discounts for this last sale.

The promoter claims a total spend of $380 for a total savings of $24.

Anyone? Moriori (talk) 23:03, 7 October 2016 (UTC)

The Accumulator first pays $280 for 140 gallons and a stack of coupons worth $8.40, then buys 50 gallons for $97 minus $8.40, so the total spending is $280 + $97 - $8.40 or $368.60 for 190 gallons, saving $11.40, not $24. The $24 makes sense only if the coupon is worth more than 6¢ if you save it for later. The $400 or $380 is the price before any discounts, not after. Your skepticism is well warranted. —Tamfang (talk) 00:14, 8 October 2016 (UTC)

Ahhhhh, so my logic wasn't so fuzzy after all, because that's what I had figured. The problem is, the oil company indicates that the discount is always 6 cents. A guy I know is totally sucked in, and won't listen to anyone who says otherwise. Cheers.Moriori (talk) 00:27, 8 October 2016 (UTC)

I can almost get their results, if I add the following assumptions (beyond gasoline costing $2 a litre):

1) For every purchase of gas, $40 or over, they get a 6 cent per litre discount coupon, which can either be used on that purchase or accumulated.

2) Up to 8 discount coupons may be used for each purchase, giving you up to 48 cents off per litre, on purchases up to $120 in gasoline.

3) Once a coupon is used it can not be used again.

Note that the assumptions might have also been defined in litres, so purchases of 20 litres or more get a coupon, which can be used for up to 60 litres. So, using these assumptions:

Redeemer buys 50 litres at $100 and uses discount of 6 cents per liter, for $3 off, each time. He does this 4 times, for 200 litres of gas purchased at $400, and a savings of $12.

Accumulator first buys 20 liters at $40 to get the 6 cents per litre coupon, which he saves. He does this 7 times, buying $280 worth of gas and getting 7 coupons. The 8th time he buys $120 worth of gas (60 litres), getting one more coupon, and uses all 8 coupons, which gives him 48 cents off per litre, times 60 litres, so $28.80 in savings.

Under both scenarios he buys $400 (200 litres) of gas, less the discount, but the discount is higher for the accumulator, because he used the coupons on larger purchases, so the discount applies to more litres.

Of course, in the real world, the accumulator's strategy may fail, for many reasons. More trips to the gas station to get those small amounts of gas may waste fuel and depreciate the car more than the value of the savings. The coupons may be lost. The promotion may end before they are redeemed. Etc. StuRat (talk) 00:39, 8 October 2016 (UTC)

Ummmm. Lets take a apples v apples scenario where both end up buying the same number of litres.

Redeemer buys 50 litres at $100 and uses discount of 6 cents per liter, for $3 off. He does this 4 times, for 200 litres of gas purchased at $400, and a total accumulated discount of $12.

Accumulator, on 5 occasions, buys 20 liters at $40 to get the 6 cents per litre discount. His accumulated discount is 5 x $1.20 = $6 which is correct for 100 litres at discount of 6 cents a litre. At his next purchase, he buys 100 litres of gas worth $200. If -- as advertised -- the discount is 6 cents a litre, then he has purchased a total of 200 litres of gas and should get a total accumulated discount of $12. No? Moriori (talk) 01:20, 8 October 2016 (UTC)

I think you and Tamfang are assuming that if you buy n liters, you get a coupon good for $0.06n off any purchase. StuRat is assuming that if you buy n liters, you get a coupon good for 3% off any purchase, that moreover can be stacked so that 8 coupons get you 24% off a single purchase. StuRat's assumption is consistent with the oil company's examples if you assume that the quoted total costs are before the discount, not after (i.e. the redeemer spends $388 and the accumulator spends $356). -- BenRG (talk) 01:57, 8 October 2016 (UTC)

Correct, except I would define it as a cents per litres discount rather than a percentage of dollars spent. So, eight 6 cents per litre coupons can be stacked, for up to 48 cents per litre off, total. StuRat (talk) 14:24, 8 October 2016 (UTC)

If this is a real-life promotion, is there a web site where we can consult the actual rules? --69.159.61.230 (talk) 08:29, 8 October 2016 (UTC)

Here's a url with a bit about it - https://www.aa.co.nz/aasmartfuel/how-aa-smartfuel-works/#tips Note the video at the top right which includes "...you can accumulate those savings to get a bigger discount per litre, meaning your fuel will be cheaper for sure, maybe even free". The "bigger discount per litre" is not consistent with their earlier illustration which quite clearly showed the price per litre is $2 and the discount is 6 cents a litre. Anyway, looks as if you do this often enough and you get free gas. Yeah right! Moriori (talk) 00:12, 10 October 2016 (UTC)

Looking through that website, it looks like the limit to which discounts can apply is 50 litres, but you can apply as many stacked discounts as you like, potentially getting that 50 litres free. However, they don't list the minimum purchase required to get each discount, or what each discount is, as apparently each retailer sets that differently. But, presumably, you need to buy lots of gas at full price to get that 50 litres "free". StuRat (talk) 00:35, 10 October 2016 (UTC)

Aha, there's a basic point that hasn't been covered: buying fuel isn't the only way for members to get the points. At the top of the linked page is a link "shop and swipe", pointing to this page. Obviously if you accumulate points by buying other stuff, then you can spend more of them when you do buy fuel. --69.159.61.230 (talk) 03:56, 10 October 2016 (UTC)