Step By Step Calculus » 17.1 - Infinite Series as a Sequence

show_hiddenExpand All [+]
Synopsis
Any convergent infinite series \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i is the limit of partial sums\displaystyle s_n=\sum_{i=1}^n a_i=a_1+a_2+\cdots+a_n \displaystyle s_n=\sum_{i=1}^n a_i=a_1+a_2+\cdots+a_n in the sense s_n\to ss_n\to s as a sequence.
If this sequence \{s_n\}\{s_n\} converges to a finite number ss, i.e. \displaystyle \lim_{n\to\infty} s_n=s\displaystyle \lim_{n\to\infty} s_n=s, the corresponding infinite series is convergent, otherwise it is divergent. When convergent, ss is called the sum of the series.
We establish



\displaystyle {\textbf{(Geometric Series Rule)}\qquad\qquad \sum_{i=1}^{\infty}ar^{i-1}=\frac{a}{1-r},\qquad \textrm{ for }|r|<1. }\displaystyle {\textbf{(Geometric Series Rule)}\qquad\qquad \sum_{i=1}^{\infty}ar^{i-1}=\frac{a}{1-r},\qquad \textrm{ for }|r|<1. }




and


(Telescoping Series Rule)
\displaystyle {\sum_{i=1}^{\infty}\frac{1}{(i+p)(i+q)}=\frac{1}{q-p}\left[\frac{1}{1+p}+\frac{1}{2+p}+\cdots+\frac{1}{1+(q-1)}\right]\;\;\textrm{ for } p<q,\textrm{ and } p,q\in\mathbb{Z}^+}\displaystyle {\sum_{i=1}^{\infty}\frac{1}{(i+p)(i+q)}=\frac{1}{q-p}\left[\frac{1}{1+p}+\frac{1}{2+p}+\cdots+\frac{1}{1+(q-1)}\right]\;\;\textrm{ for } p<q,\textrm{ and } p,q\in\mathbb{Z}^+}


Equivalent Series Theorem: Consider two series \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i and \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i. If \displaystyle \sum_{i=1}^{\infty} (a_i-b_i)\displaystyle \sum_{i=1}^{\infty} (a_i-b_i) is convergent, then either both series are convergent or both series are divergent, i.e., both series are equivalent.
From that one can infer the following key property: If \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i, \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i are equivalent and \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i, \displaystyle \sum_{i=1}^{\infty} c_i\displaystyle \sum_{i=1}^{\infty} c_i are equivalent, then \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i, \displaystyle \sum_{i=1}^{\infty} c_i\displaystyle \sum_{i=1}^{\infty} c_i are equivalent.
If the series \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i is convergent, then we must have \displaystyle \lim_{n\to \infty} a_n=0\displaystyle \lim_{n\to \infty} a_n=0. This gives rise to the following divergence test.
(\mathbf{n^{\textrm{th}}}\mathbf{n^{\textrm{th}}}-Term Divergence Test)
\displaystyle { \textit{If $\displaystyle \lim_{n\to \infty} a_n$ does not exist or $\displaystyle \lim_{n\to \infty} a_n\neq 0$, then the series $\displaystyle \sum_{i=1}^{\infty} a_i$ diverges. }}\displaystyle { \textit{If $\displaystyle \lim_{n\to \infty} a_n$ does not exist or $\displaystyle \lim_{n\to \infty} a_n\neq 0$, then the series $\displaystyle \sum_{i=1}^{\infty} a_i$ diverges. }}


Following are some properties of series.
  • Suppose both \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i and \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i is convergent and cc is a constant. Then
    \displaystyle \sum_{i=1}^{\infty }(ca_{i}+ b_{i})=\displaystyle c\sum_{i=1}^{\infty }a_{i}+ \displaystyle \sum_{i=1}^{\infty }b_{i}\displaystyle \sum_{i=1}^{\infty }(ca_{i}+ b_{i})=\displaystyle c\sum_{i=1}^{\infty }a_{i}+ \displaystyle \sum_{i=1}^{\infty }b_{i}.
  • if \displaystyle \sum_{i=1}^{\infty} a_i\displaystyle \sum_{i=1}^{\infty} a_i is divergent to \infty\infty and \displaystyle \sum_{i=1}^{\infty} b_i\displaystyle \sum_{i=1}^{\infty} b_i is convergent and cc is a constant. Then
    \displaystyle \sum_{i=1}^{\infty} (ca_i + b_i)= \left\{\begin{array}{ll} \infty & \textrm{ if }c>0 \\ -\infty & \textrm{ if } c<0\\ \sum_{i=1}^{\infty} b_i &\textrm{ if } c=0\end{array}\right. \displaystyle \sum_{i=1}^{\infty} (ca_i + b_i)= \left\{\begin{array}{ll} \infty & \textrm{ if }c>0 \\ -\infty & \textrm{ if } c<0\\ \sum_{i=1}^{\infty} b_i &\textrm{ if } c=0\end{array}\right.