Show that the sequence, $E_{n=0}^∞ (\exp\{-(E_n)/((k_B)T)\})$ is convergent and find its sum.
I usually know how to do this with other functions, but I feel kind of lost in this one since it is an exponential function with different variables?
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Do you mean $\sum_{n=0}^\infty$? What is $E_n$? – Robert Israel Nov 24 '19 at 15:28
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Yes I do, it says that E_n is the energy. It says E_n = E_0 +2nE_0 where E_0 is a constant, and k_B is boltzman's constant = 0.67 – Lioness Queen Nov 24 '19 at 15:31
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Are $E_0$ and $T$ positive? Hint: geometric series. – Robert Israel Nov 24 '19 at 16:03
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I don't really know, I suppose T (temperature) would be positive, but I had guessed it would be a geometric series, but I can't figure out how to come to any conclusion. I guess one could assume that both the energy and temperature would be positive, but then we would have exponetial to something negative? – Lioness Queen Nov 24 '19 at 16:45
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Let $e^{\frac{-E_0}{k_B T}} = r$. Note that $r<1$.
Then \begin{align} \sum_0^{\infty} e^{\frac{-E_0(1+2n)}{k_B T}} &= \sum_0^{\infty} r^{1+2n} \\ &= r\sum_0^{\infty} r^{2n} \\ &= \frac{r}{1-r^2} \\ &= \frac{e^{\frac{-E_0}{k_B T}}}{1-e^{\frac{-2E_0}{k_B T}}} \\ \end{align}
Ishan Deo
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