Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/35

 

CHAPTER III

 

THEORY OF LIMITS

 

13. Limit of a variable. If a variable $\scriptstyle {v}$ takes on successively a series of values that approach nearer and nearer to a constant value $\scriptstyle {l}$ in such a manner that $\scriptstyle {|v-l|}$^[1] becomes and remains less than any assigned arbitrarily small positive quantity, then $\scriptstyle {v}$ is said to approach the limit $\scriptstyle {l}$, or to converge to the limit $\scriptstyle {l}$. Symbolically this is written

$\scriptstyle {\mathbf {limit~} v=l,\mathbf {~or,~} v\doteq l}$.

The following familiar examples illustrate what is meant:

(1) As the number of sides of a regular inscribed polygon is indefinitely increased, the limit of the area of the polygon is the area of the circle. In this case the variable is always less than its limit.

(2) Similarly, the limit of the area of the circumscribed polygon is also the area of the circle, but now the variable is always greater than its limit.

(3) Consider the series

The sum of any even number $\scriptstyle {(2n)}$ of the first terms of this series is

(B)

${\begin{aligned}&\scriptstyle {S_{2n}=1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{8}}+\cdots +{\frac {1}{2^{2n-2}}}-{\frac {1}{2^{2n-1}}},}\\&\scriptstyle {S_{2n}={\frac {{\frac {1}{2^{2n}}}-1}{-{\frac {1}{2}}-1}}={\frac {2}{3}}-{\frac {1}{3\cdot 2^{2n-1}}}.}\end{aligned}}$

By 6, p. 1

Similarly, the sum of any odd number $\scriptstyle {(2n+1)}$ of the first terms of the series is

(C)

${\begin{aligned}&\scriptstyle {S_{2n+1}=1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{8}}+\cdots -{\frac {1}{2^{2n-1}}}+{\frac {1}{2^{2n}}},}\\&\scriptstyle {S_{2n+1}={\frac {-{\frac {1}{2^{2n+1}}}-1}{-{\frac {1}{2}}-1}}={\frac {2}{3}}+{\frac {1}{3\cdot 2^{2n}}}.}\end{aligned}}$

By 6, p. 1

1. To be read the numerical value of the difference between $\scriptstyle {v}$ and $\scriptstyle {l}$.

11