# proof of limit comparison test

The main theorem we will use is the comparison test^{}, which basically states that if ${a}_{n}>0$, ${b}_{n}>0$ and there is an $N$ such that for all $n>N$, $$ , then if ${\sum}_{i=1}^{\mathrm{\infty}}{b}_{n}$ converges^{} so will ${\sum}_{i=1}^{\mathrm{\infty}}{a}_{n}$.

Suppose ${lim}_{n\to \mathrm{\infty}}\frac{{a}_{n}}{{b}_{n}}=L$ where $L$ can be a non negative real number or $+\mathrm{\infty}$.

By definition, for $L$ finite, this means that for every $\u03f5>0$ there is a natural number^{} ${n}_{\u03f5}$ such that for all $n>{n}_{\u03f5}$, $$

To make matters more concrete choose $\u03f5=\frac{L}{2}$ and assume $L\ne 0$ and finite.

$$, for all $n>{n}_{\frac{L}{2}}$.

If ${\sum}_{i=1}^{\mathrm{\infty}}{b}_{n}$ converges, so will ${\sum}_{i=1}^{\mathrm{\infty}}\frac{3L}{2}{b}_{n}$ and thus by the comparison test, ${\sum}_{i=1}^{\mathrm{\infty}}{a}_{n}$ will also be convergent^{}.

For the reverse result, consider ${lim}_{n\to \mathrm{\infty}}\frac{{b}_{n}}{{a}_{n}}=\frac{1}{L}$, since if $L$ is finite so will $\frac{1}{L}$, applying the previous result we can say that if ${\sum}_{i=1}^{\mathrm{\infty}}{a}_{n}$ converges so will ${\sum}_{i=1}^{\mathrm{\infty}}{b}_{n}$

Consider the case $L=0$, clearly $L={0}^{+}$ since both ${a}_{n}$ and ${b}_{n}$ are positive, this means that for all $\u03f5>0$ there exists ${n}_{\u03f5}$ such that for all $n>{n}_{\u03f5}$, $$.

Considering $\u03f5=1$ we get the exact formulation of the comparison test, so if ${\sum}_{i=1}^{\mathrm{\infty}}{b}_{n}$ converges so will ${\sum}_{i=1}^{\mathrm{\infty}}{a}_{n}$.

For the case $L=+\mathrm{\infty}$ just apply the result to ${lim}_{n\to \mathrm{\infty}}\frac{{b}_{n}}{{a}_{n}}=0$ to conclude that if ${\sum}_{i=1}^{\mathrm{\infty}}{a}_{n}$ converges so will ${\sum}_{i=1}^{\mathrm{\infty}}{b}_{n}$

Title | proof of limit comparison test |
---|---|

Canonical name | ProofOfLimitComparisonTest |

Date of creation | 2013-03-22 15:35:54 |

Last modified on | 2013-03-22 15:35:54 |

Owner | cvalente (11260) |

Last modified by | cvalente (11260) |

Numerical id | 4 |

Author | cvalente (11260) |

Entry type | Proof |

Classification | msc 40-00 |