Series help

Hello, and welcome to FMH! :)

Let's write:

[MATH]S=\sum_{k=-1}^{\infty}\left(e^{-k}\right)[/MATH]
Now suppose we multiply both sides by \(e\):

[MATH]Se=\sum_{k=-1}^{\infty}\left(e^{-k+1}\right)=\sum_{k=-2}^{\infty}\left(e^{-k}\right)[/MATH]
What do you get when you subtract the latter from the former?
 
Suppose we write:

[MATH]Se=e^2+\sum_{k=-1}^{\infty}\left(e^{-k}\right)[/MATH]
Now, when we subtract the former from the latter (I meant to suggest that in my first post, but either way works) we get

[MATH]Se-S=e^2+\sum_{k=-1}^{\infty}\left(e^{-k}\right)-\sum_{k=-1}^{\infty}\left(e^{-k}\right)=e^2[/MATH]
Now solve for \(S\)...what do you find?
 
Evaluate the infinite sum e+e^0+e^-1+e^-2+e^-3+...
I don't think you've told us anything about what you do or don't understand in these problems, which makes it hard to know what kind of answer to give. Do you know sigma notation? Do you know formulas for geometric series? In the latter case, you could use that directly here. (MarkFL is showing you how to derive it in this special case.)
 
I went about it differently but I don’t think I did it right. I used the formula for sum of an infinite geometric series so I said a=e, r=1/e, did e/1-1/e and got 4.300258535. So I’m confused if I did it the wrong way and it’s wrong.
 
You should have gotten 4.30025853532837. You are just rounding sooner than I.

It is hoped that you could produce the exact value. NO ROUNDING! [math]\dfrac{e}{1-e^{-1}}[/math]
 
I went about it differently but I don’t think I did it right. I used the formula for sum of an infinite geometric series so I said a=e, r=1/e, did e/1-1/e and got 4.300258535. So I’m confused if I did it the wrong way and it’s wrong.
Is there a reason you don't trust your work? There may be something you need to discuss.

Note that what you meant is e/(1-1/e); the parentheses are essential!
 
k
Are

It is hoped that you could produce the exact value. NO ROUNDING! [math]\dfrac{e}{1-e^{-1}}[/math]



steph7, if there is an answer key with the exact answer, it may have used
positive exponents only. Take the answer in this quote box and multiply
the numerator and the denominator each by e:

\(\displaystyle \dfrac{e^2}{e \ - \ 1}\)
 
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