Definition of convergence of a series

Question

My text book gives the following definition of convergence of a series:

A sequence of real numbers is said to be convergent if it has a finite real number $L$ as its limit. We then say that the sequence $x$ converges to $L$.

But I find it to be confusing, as from the above definition of convergence, $1/n$ turns out to be a Cauchy sequence. But wikipedia seems to be providing a different definition of convergence Definition of Convergent Series

PS: I am just a newbie in real analysis, so I do request you to be a little more elaborative.

Edit: Oh, that question was stupid enough. I really did mix up the two terms "sequence" and "series". Thanks for pointing this out.

Answer 1

There are two different kinds of objects that you are studying: sequences, and series.

A sequence $(a_n)$ of real numbers converges if there is some finite real number $L$ such that $\lim_{n\to\infty} a_n = L$. What this means is that we can make the difference between $a_n$ and $L$ as small as we like by choosing a number $n$ that is large enough. More formally,

A sequence $(a_n)$ converges to $L$ if for any $\varepsilon > 0$ there exists some $N$ so large that $n \ge N$ implies that $|a_n - L| < \varepsilon$.

A series $\sum_{n=1}^{\infty} a_n$ converges if the sequence of partial sums $S_N$ converges, where $$ S_N := \sum_{n=1}^{N} a_n. $$ That is, in order to discuss the convergence of a series, we first turn the series into a sequence, then seek to understand the properties of that sequence. Thus a series is said to converge to a limit $S$ if the sequence $(S_N)$ (as defined above) converges to $S$ as a sequence. In notation, we might write $$ \sum_{n=1}^{\infty} a_n = S \iff \lim_{N\to\infty} S_n = \lim_{N\to\infty} \left( \sum_{n=1}^{N} a_n \right) = S.$$

The classic example (cited by other responses to your question) is the harmonic series, $$ \sum_{n=1}^{\infty} \frac{1}{n}. $$ The individual terms $\frac{1}{n} \to 0$ as $n\to \infty$, but the series does not converge. To see this, write $$ \frac{1}{2} + \left( \frac{1}{3} + \frac{1}{4} \right) + \left( \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} \right) + \dotsb.$$ Notice that $\frac{1}{3} > \frac{1}{4}$, thus $$ \frac{1}{3} + \frac{1}{4} > \frac{1}{4} + \frac{1}{4} = \frac{1}{2}. $$ Similarly, each term in the next group is bigger than $\frac{1}{8}$, and so $$ \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} > \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{1}{2}. $$ By a similar process, we can continue to lump terms together to add up to $\frac{1}{2}$, which gives $$ \sum_{n=1}^{\infty} \frac{1}{n} = 1+ \frac{1}{2} + \left( \frac{1}{3} + \frac{1}{4} \right) + \left( \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} \right) + \dotsb \ge \sum_{k=1}^{\infty} \frac{1}{2} = \infty, $$ where the inequality can be justified via the Squeeze Theorem.

Answer 2

One of them is the definition of convergent sequence. The other one is the definition of convergent series. Of course they are different. How could they be equal? We are talking about different concepts here.

Answer 3

The sequence $(\frac{1}{n})_{n=1}^\infty$ converges, but the sequence $(\sum_{n=1}^N \frac{1}{n})_{N=1}^\infty$ does not. When your book writes "as its limit", it means in the latter form. That is, we define $$ \sum_{n=1}^\infty a_n := \lim_{N\to\infty}\sum_{n=1}^Na_n,$$ provided the limit exists.

Answer 4

It looks like you may be confusing a sequence with a series.

In a sequence you are looking at a list of terms. With a series you are adding those terms.

The sequence $\{\frac1n\}$ converges to $0$.

The series $\sum_1^\infty \frac1n$ diverges.