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Real Limits
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Real Series


We want to sum infinitely many real numbers.


Let $(a_k)_{k \in \N }$ be a sequence in $\R $. Define $(s_n)_{n \in \N }$ by

\[ s_n = \sum_{k = 1}^{n} a_k. \]

We call $s_n$ the $n$th partial sum of $(x_k)$. In other words, the first partial sum $s_1$ is $a_1$, the second partial sum $s_2$ is $a_1 + a_2$, the third partial sum $s_3$ is $a_1 + a_2 + a_3$ and so on.

We call $(s_n)$ the sequence of partial sums or series of $(a_k)$. If the series converges, then we say that $(a_k)$ is summable. Clearly not every series is summable: consider, for example, $a_k = 1$ for all $k$. It has the divergent series $(1, 2, 3, 4, 5, \dots )$.


If the sequence is summable, then there exists a unique $s \in \R $ (the limit), which we denote

\[ s = \lim_{n\to\infty} s_n = \lim_{n\to\infty} \sum_{k = 1}^{n} a_k. \]

We read these relations aloud as “s is the limit as n goes to infinity of s n” and “s is the limit as n goes to infinity of the sum of a k from k equals 1 to n.” We often avoid referencing $s_n$ by abbreviating the above with

\[ \sum_{k = 1}^{\infty} a_k = s. \]

We read this notation aloud as “the sum from 1 to infinity of a k is s.” The notation is subtle, and requires justification by the algebra of series.1


For a series to converge, intuition suggests that the additional terms added should be getting smaller and smaller. Indeed:

Let $(a_k)_{k \in \N }$ be a sequence of real numbers. If $(a_k)$ is summable then $a_k$ converges to $0$.2

The converse of this theorem has immediate relevance as a preliminary test for determining whether a series converges.

If $(a_k)$ does not converge or converges to $a_0 \neq 0$, then it is not summable.

  1. Future editions will include such justification. ↩︎
  2. Future editions will include an account. ↩︎
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