Editing Telescoping series (section)

== Applications ==
In [[probability theory]], a [[Poisson process]] is a stochastic process of which the simplest case involves "occurrences" at random times, the waiting time until the next occurrence having a [[memorylessness|memoryless]] [[exponential distribution]], and the number of "occurrences" in any time interval having a [[Poisson distribution]] whose expected value is proportional to the length of the time interval.  Let ''X''<sub>''t''</sub> be the number of "occurrences" before time ''t'', and let ''T''<sub>''x''</sub> be the waiting time until the ''x''th "occurrence".  We seek the [[probability density function]] of the [[random variable]] ''T''<sub>''x''</sub>.  We use the [[probability mass function]] for the Poisson distribution, which tells us that
: <math> \Pr(X_t = x) = \frac{(\lambda t)^x e^{-\lambda t}}{x!}, </math>
where λ is the average number of occurrences in any time interval of length 1.  Observe that the event {''X''<sub>''t''</sub> ≥ x} is the same as the event {''T''<sub>''x''</sub> ≤ ''t''}, and thus they have the same probability. Intuitively, if something occurs at least <math>x</math> times before time <math>t</math>, we have to wait at most <math>t</math> for the <math>xth</math> occurrence.  The density function we seek is therefore
: <math>
\begin{align}
f(t) & {} = \frac{d}{dt}\Pr(T_x \le t) = \frac{d}{dt}\Pr(X_t \ge x) = \frac{d}{dt}(1 - \Pr(X_t \le x-1)) \\  \\
& {} =  \frac{d}{dt}\left( 1 - \sum_{u=0}^{x-1} \Pr(X_t = u)\right)
= \frac{d}{dt}\left( 1 - \sum_{u=0}^{x-1} \frac{(\lambda t)^u e^{-\lambda t}}{u!}  \right) \\  \\
& {} = \lambda e^{-\lambda t} - e^{-\lambda t} \sum_{u=1}^{x-1} \left( \frac{\lambda^ut^{u-1}}{(u-1)!} - \frac{\lambda^{u+1} t^u}{u!} \right)
\end{align}
</math>
The sum telescopes, leaving
: <math> f(t) = \frac{\lambda^x t^{x-1} e^{-\lambda t}}{(x-1)!}. </math>

For other applications, see:
* [[Proof that the sum of the reciprocals of the primes diverges]], where one of the proofs uses a telescoping sum;
* [[Fundamental theorem of calculus]], a continuous analog of telescoping series;
* [[Order statistic]], where a telescoping sum occurs in the derivation of a probability density function;
* [[Lefschetz fixed-point theorem]], where a telescoping sum arises in [[algebraic topology]];
* [[Homology theory]], again in algebraic topology;
* [[Eilenberg–Mazur swindle]], where a telescoping sum of knots occurs;
* [[Faddeev–LeVerrier algorithm]].