section_path: Basic properties/Convergent series
section_name: Convergent series
paragraph_text: Convergent series
  
  A series Σ a is said to converge or to be convergent when the sequence (s) of partial sums has a finite limit. If the limit of s is infinite or does not exist, the series is said to diverge. When the limit of partial sums exists, it is called the value (or sum) of the series
  
  ∑ n = 0 ∞ a n = lim k → ∞ s k = lim k → ∞ ∑ n = 0 k a n. {\displaystyle \sum _{n=0}^{\infty }a_{n}=\lim _{k\to \infty }s_{k}=\lim _{k\to \infty }\sum _{n=0}^{k}a_{n}.}
  
  An easy way that an infinite series can converge is if all the a are zero for n sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.
  
  Working out the properties of the series that converge, even if infinitely many terms are nonzero, is the essence of the study of series. Consider the example
  
  1 + 1 2 + 1 4 + 1 8 + ⋯ + 1 2 n + ⋯. {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n}}}+\cdots.}
  
  It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: When we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2. In other words, the series has an upper bound. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted S, it can be seen that
  
  S / 2 = 1 + 1 2 + 1 4 + 1 8 + ⋯ 2 = 1 2 + 1 4 + 1 8 + 1 16 + ⋯. {\displaystyle S/2={\frac {1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots }{2}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots.}
  
  Therefore,
  
  S − S / 2 = 1 ⇒ S = 2. {\displaystyle S-S/2=1\Rightarrow S=2.}
  
  The idiom can be extended to other, equivalent notions of series. For instance, a recurring decimal, as in
  
  x = 0.111 …, {\displaystyle x=0.111\dots,}
  
  encodes the series
  
  ∑ n = 1 ∞ 1 10 n. {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{10^{n}}}.}
  
  Since these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that 9 × 0.111... = 0.999... = 1, which only relies on the fact that the limit laws for series preserve the arithmetic operations; for more detail on this argument, see 0.999....