Infinitesimals And Limits
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[i] Hence ^I< I X ' e. < I X ' ^3< I X ' etc < etc. , ^«< I X ' Cj €, €3 .... €„< I x„ INFINITESIMAI. S AND LIMITS 1 3 Now making x increase indefinitely we can make — an infinitesimal. Hence by [i] c^ f..^ S • • • ^n is an infin- itesimal. Q. E. D. PROPOSITION 5 14. First. — A power of an infinitesimal is itself an infinitesimal if its exp orient is a positive finite other than o. Second. — // is an infinite if its exponent is a negative finite other than o. Third. — // is equal to i if its exponent is o. First. Let a be any infinitesimal and p a positive finite other than o. [i] Then a^ = a a a a ... To / factors. [2] But a a a ... To / factors is an infinitesi- mal, by § 13. [3] Hence a^ ^^ an infinitesimal. Second. Let — / be a negative finite other than o. [4] Then ''~'=^' But since a is an infinitesimal [5] a^ = an infinitesimal, by case i. by § 10. [6] and I = an infinite. Hence by [4] we get [7] a^ — an Third. infinite. Let P = 0. Then a. ' . = a« = = I.
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To quickly assess the difficulty of the text, read a short excerpt:
[i] Hence ^I< I X ' e. < I X ' ^3< I X ' etc < etc. , ^«< I X ' Cj €, €3 .... €„< I x„ INFINITESIMAI. S AND LIMITS 1 3 Now making x increase indefinitely we can make — an infinitesimal. Hence by [i] c^ f..^ S • • • ^n is an infin- itesimal. Q. E. D. PROPOSITION 5 14. First. — A power of an infinitesimal is itself an infinitesimal if its exp orient is a positive finite other than o. Second. — // is an infinite if its exponent is a negative finite other than o. Third. — // is equal to i if its exponent is o. First. Let a be any infinitesimal and p a positive finite other than o. [i] Then a^ = a a a a ... To / factors. [2] But a a a ... To / factors is an infinitesi- mal, by § 13. [3] Hence a^ ^^ an infinitesimal. Second. Let — / be a negative finite other than o. [4] Then ''~'=^' But since a is an infinitesimal [5] a^ = an infinitesimal, by case i. by § 10. [6] and I = an infinite. Hence by [4] we get [7] a^ — an Third. infinite. Let P = 0. Then a. ' . = a« = = I.
What to read after Infinitesimals And Limits?
You can find similar books in the "Read Also" column, or choose other free books by Joseph Johnston Hardy to read online
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