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I found this problem on summations, and I'm not really sure how to solve it. Could someone give a hint as to how to do so? Find the value of

$$\sum_{i=1}^{1000}f\left(\frac{i}{1000}\right),\qquad f(x) = \frac{4^x}{4^x+2}$$ It came on an exam where we couldn't use calculators, and it apparently is an integer answer, though Wolfram Alpha disagrees...(Even if it isn't, I would still like to know how to do it)

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    what is this $$\sum_{i=1}^{1000}=$$? – 2017-01-31
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    Umm... I meant to say the summation of the thing? For i=1 to 1000? – 2017-01-31
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    But what is the _thing_? – 2017-01-31
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    What is the _summand_? Do you have definition of _f(n)_? – 2017-01-31
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    Use If $f(n)=\dfrac{4^n}{4^n+2}$ $$f(1-n)=\dfrac{4^{1-n}}{4^{1-n}+2}=\dfrac4{4+2\cdot4^n}=\dfrac2{2+4^n}=1-f(n)$$ – 2017-01-31
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    I'm sorry about that, not really familiar with the formal notation. What i meant to say is, if f(n)=4^n/(4^n)+2, find the sum of the values of it as n ranges from 1/1000 to 1 – 2017-01-31
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    Wow! Thanks a lot, that helped. So, everything cancels out to give 500 then? – 2017-01-31
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    @AnishHebbar: It would if $i = 0$ were the lower limit. As it is, I think you're missing a $1/3$ term. – 2017-01-31
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    Yeah, noticed that. My bad! – 2017-01-31

2 Answers 2

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We have $$ f(x)+f(1-x) = \frac{4^x}{4^x+2}+\frac{4^{1-x}}{4^{1-x}+2} = \frac{4^x}{4^x+2}+\frac{2}{4^x+2}=1 $$ hence $$ \left[f\left(\frac{1}{1000}\right)+f\left(\frac{999}{1000}\right)\right]+\ldots+\left[f\left(\frac{499}{1000}\right)+f\left(\frac{501}{1000}\right)\right]=499 $$ and your sum is just $499+f(1)+f\left(\frac{1}{2}\right)=\color{red}{500+\frac{1}{6}}$.

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    @JackDAurizio: Nice answer! (+1) – 2017-02-01
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$$\sum_{n=1}^{1000}\frac{4^n}{4^n+2} = \sum_{n=1}^{1000}\frac{2^{2n}}{2^{2n}+2} = \sum_{n=1}^{1000}\frac{2^{2n-1}}{2^{2n-1}+1} = \frac23 + \frac89 + \frac{32}{33} + \frac{128}{129} + \cdots + \frac{2^{1999}}{{2^{1999}+1}} \approx 999.5149000482058.$$

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    But this answers a different question-- I summed over 1 to 1000 instead of 1/1000 to 1. Should I just delete the post in a situation like this? – 2017-01-31
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    I wouldn't delete it, but I would add something in it to state that the question you answered was different (due to a lack of clarity in the question or a misunderstanding by you). – 2017-01-31