Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/236

CHAPTER XVII

SERIES

134. Introduction. A series is a succession of separate numbers which is formed according to some rule or law. Each number is called a term of the series. Thus

$1,\ 2,\ 4,\ 8,\ ...,\ 2^{n-1}$

is a series whose law of formation is that each term after the first is found by multiplying the preceding term by 2; hence we may write down as many more terms of the series as we please, and any particular term of the series may be found by substituting the number of that term in the series for $n$ in the expression $2^{n-1}$, which is called the general or nth term of the series.

EXAMPLES

In the following six series:

(a) Discover by inspection the law of formation;

(b) write down several terms more in each;

(c) find the nth or general term.

Series	nth term
1. $1,\,3,\,9,\,27,\,\cdots ,$	$\ 3^{n-1}$.
2. $-a,\,+a^{2},\,-a^{3},\,+a^{4},\,\cdots ,$	$\ \left(-a\right)^{n}$.
3. $1,\,4,\,9,\,16,\,\cdots ,$	$\ n^{2}$.
4. $x,\,{\frac {x^{2}}{2}},\,{\frac {x^{3}}{3}},\,{\frac {x^{4}}{4}},\,\cdots ,$	$\ {\frac {x^{n}}{n}}$.
5. $4,\,-2,\,+1,\,-{\frac {1}{2}},\,\cdots ,$	$\ 4\left(-{\frac {1}{2}}\right)^{n-1}$.
6. ${\frac {3y}{2}},\,{\frac {5y^{2}}{5}},\,{\frac {7y^{3}}{10}},\,\cdots ,$	$\ {\frac {2n+1}{n^{2}+1}}y^{n}$.

Write down the first four terms of each series whose nth or general term is given below :

nth term	Series
7. $n^{2}x^{n}$.	$x,\,4x^{2},\,9x^{3},\,16x^{4}$.
8. ${\frac {x^{n}}{1+{\sqrt {n}}}}$.	${\frac {x}{2}},\,{\frac {x^{2}}{1+{\sqrt {2}}}},\,{\frac {x^{3}}{1+{\sqrt {3}}}},\,{\frac {x^{4}}{1+{\sqrt {4}}}}$.
9. ${\frac {n+2}{n^{3}+1}}$.	${\frac {3}{2}},\,{\frac {4}{9}},\,{\frac {5}{28}},\,{\frac {6}{65}}$.