2.11 Infinite Series

Definition 2.11.1.

Let \(\{a_n\}^{\infty }_{n=1}\) be an infinite sequence. An infinite is given by \[\sum ^{\infty }_{n=1}a_n = a_1 + a_2 + \cdots \cdots \cdots \]

The partial sums of the series are given \[S_1 = a_1\hspace {0.2cm} , \hspace {0.2cm} S_2 = a_1 + a_2\hspace {0.2cm} , \hspace {0.2cm} S_3 = a_1 + a_2 + a_3\hspace {0.3cm} \cdots \cdots \cdots \hspace {0.3cm} S_N = a_1 + a_2 + \cdots \cdots a_N = \sum ^n_{n=1}\]

Now \(\displaystyle {S_n = \sum ^n_{k=1}a_k}\) is called the \(n^{\text {th}}\) partial sum.

Definition 2.11.2.

If \(\lim \limits _{n \rightarrow \infty } S_n = S\), a finite number then the series \(\displaystyle {\sum ^{\infty }_{n=1}a_n}\hspace {0.3cm}\) is said to converge to \(s\) i.e \[\sum ^{\infty }_{n=1}a_n = a_1 + a_2 + \cdots \cdots \cdots = S\]

If the \(\lim \limits _{n \rightarrow \infty } S_n \) does not exist, then the series \(\displaystyle {\sum ^{\infty }_{n=1}a_n}\) is said to be divergent.

Properties of Series

1.: A convergent(divergent) series remains convergent(divergent) after any or all its terms are altered.
2.: The sum of a convergent series is unique.
3.: If the series \(\sum a_n = S\) then \(\sum K a_n = KS\), \(\hspace {0.2cm}K\) being a constant. If \(\sum a_n\) diverges, so also \(\sum Ka_n\) diverges, if \(K\neq 0\).
4.: If \(\sum a_n\) converges, the \(\displaystyle {\lim \limits _{n \rightarrow \infty } a_n = 0}\). The converse of this property is not true, for example for the series \(\displaystyle {\sum ^{\infty }_{n =1}\frac {1}{n}}\) \[S_1 = 1\hspace {0.2cm} , \hspace {0.2cm} S_2 = 1 + \frac {1}{2} = \frac {3}{2}\hspace {0.2cm},\hspace {0.2cm} S_3 = 1 + \frac {1}{2}+ \frac {1}{3} = \frac {11}{6}\hspace {0.2cm}\cdots \cdots \cdots \cdots \] and \(\{S_n\}\) is non-decreasing and is not bounded above. Therefore \(\displaystyle {\sum ^{\infty }_{n = 1}\frac {1}{n}}\) diverges though \(\displaystyle {\lim \limits _{n \rightarrow \infty } \frac {1}{n} = 0}\)
5.: If \(\lim \limits _{n \rightarrow \infty } a_n \neq 0\), then \(\sum a_n\) diverges.
The converse is not true. For example \(\displaystyle {\sum ^{\infty }_{n=1}\frac {1}{n}}\) diverges but \(\lim \limits _{n \rightarrow \infty } \dfrac {1}{n}=0\)
6.: A positive series \(\sum a_n\) is convergent if the sequence of partial series \(\{S_n\}\) is bounded.

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