Series help
- Thread starter steph7
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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?
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?
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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?
[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?
Dr.Peterson
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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.)
Dr.Peterson
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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
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:
\(\displaystyle \dfrac{e^2}{e \ - \ 1}\)
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:
\(\displaystyle \dfrac{e^2}{e \ - \ 1}\)