_Posted on May 12, 2024_ ##### Introduction A series is an infinite sum of terms in a given sequence $(a_n)$. Notationally, $\sum_{n=m}^\infty a_n$ However, it is conceptually difficult to assign a value to such an object. If the terms in the sequence $(a_n)$ are increasing, it is intuitive that the value of $\sum_{n=m}^\infty a_n$ might go on to infinity. But for still positive sequences, are there any series which have definite values? --- ##### Zeno's Dichotomy Paradox Consider [Zeno's](https://en.wikipedia.org/wiki/Zeno_of_Elea) dichotomy paradox: Suppose [Atalanta](https://en.wikipedia.org/wiki/Atalanta) is trying to get to a particular location. Lets normalize the distance from Atalana to that location so that it is a single unit. Atalanta thus starts one unit away from the target location, then starts moving. To get there, she must travel half the distance, then half that remaining distance, then half again, and so on. Continuing in this way, she may never reach her destination, always travelling half the distance in an infinite regress. However, we know experientially that Atalanta must eventually reach her destination, right? So, we have some idea that the total distance covered will eventually reach one unit! Consider a the sequence of distances covered in this case as $(d_n)$, where $\displaystyle d_n = \frac{1}{2^n}$. We might want the associated series $\sum_{n=1}^\infty d_n$ to _equal_ $1$. --- ##### Defining series convergence Thus we justify some definition for the sentence "$\sum_{n=m}^\infty a_n = Squot;. However, we notice that when adding a finite number of such terms in $(d_n)$ they never really reach 1 but approach it arbitrarily. We thus say that $\sum_{n=m}^\infty a_n = S$ really means that the series approaches $S$, or converges to $S$. Now what does it mean, formally, for a series to converge to a value $S$? This simply means that as the upper limit of the sum increases, the value of this sum gets arbitrarily closer to $S$. Notice that there is a new sequence here, that is, $(\sum_{k=m}^na_k)$. So, as $n$ increases, we want the sequence $(\sum_{k=m}^na_k)$ to get arbitrarily close to $S$. Thus the sequence $(\sum_{k=m}^na_k)$ should approach, or converge to $S$. This is in fact precicely the typical definition of convergence for a series: that its sequence of partial sums $(\sum_{k=m}^na_k)$ [[Convergent Sequences are bounded|converges]] to $S,$ or $\displaystyle\lim_{n\rightarrow \infty}\sum_{k=m}^na_k = S$. We now have powerful machinery towards making some definite statements about the behavior of particular limits. Thus, in a future post, we shall demonstrate formally that $\sum_{n=1}^\infty\frac{1}{2^n}$ in fact converges to $1$.