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I have a set of data that I've been working with that seems to be defined by the sum of a set of exponential functions of the form [itex](1-e^{\frac{-t}{\tau}})[/itex]. I've come up with the following series which is the product of a decay function and an exponential with an increasing time constant. If this series converges (I think it does), it would be very helpful to have a single, continuous function that describes the same curve.
[itex]
f(t) = \sum_{i=1}^\infty (e^{1-i}-e^{-i})(1-exp( \frac{-t}{\tau R^i}))
[/itex]
where [itex]f[/itex] is a function of [itex]t[/itex] (time),
[itex]\tau[/itex] is a constant where [itex]\tau>0[/itex],
and [itex]R[/itex] is a constant where [itex]R>1[/itex].
Note that [itex]\lim_{t\rightarrow \infty}{f(t)=1}[/itex]
The [itex]n^{th}[/itex] term test does not reveal anything since [itex]\lim_{n\rightarrow +\infty}{f(t)}=0[/itex]
Writing out the [itex]n^{th}[/itex] partial sum doesn't reveal much to me.
Beyond that, I'm not really sure where to go.
I've plotted an example set with a [itex]n=2[/itex] partial sum and a [itex]n=10[/itex] partial sum (left column). You can ignore their comparisons to a simulation on the right.
https://photos-5.dropbox.com/t/2/AADR6nipgtsqQOHrGDcNtxiqO6N_4N6kaoNmk1NMiXTb-w/12/272245/png/32x32/1/_/1/2/exponentials_18250.png/EPrZNBjh4avTBCACKAI/Lew1XtsiNm8kaz3Vx1x-gEyH4EiHRQ5HufHxDkmFd-E?size=1280x960&size_mode=3
Can anyone help me 1)determine if this series converges and if so, 2)what the continuous function that describes the series is?
[itex]
f(t) = \sum_{i=1}^\infty (e^{1-i}-e^{-i})(1-exp( \frac{-t}{\tau R^i}))
[/itex]
where [itex]f[/itex] is a function of [itex]t[/itex] (time),
[itex]\tau[/itex] is a constant where [itex]\tau>0[/itex],
and [itex]R[/itex] is a constant where [itex]R>1[/itex].
Note that [itex]\lim_{t\rightarrow \infty}{f(t)=1}[/itex]
The [itex]n^{th}[/itex] term test does not reveal anything since [itex]\lim_{n\rightarrow +\infty}{f(t)}=0[/itex]
Writing out the [itex]n^{th}[/itex] partial sum doesn't reveal much to me.
Beyond that, I'm not really sure where to go.
I've plotted an example set with a [itex]n=2[/itex] partial sum and a [itex]n=10[/itex] partial sum (left column). You can ignore their comparisons to a simulation on the right.
https://photos-5.dropbox.com/t/2/AADR6nipgtsqQOHrGDcNtxiqO6N_4N6kaoNmk1NMiXTb-w/12/272245/png/32x32/1/_/1/2/exponentials_18250.png/EPrZNBjh4avTBCACKAI/Lew1XtsiNm8kaz3Vx1x-gEyH4EiHRQ5HufHxDkmFd-E?size=1280x960&size_mode=3
Can anyone help me 1)determine if this series converges and if so, 2)what the continuous function that describes the series is?
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