Elements of the Integral Calculus, With a Key to the Solution of Differential Equatons, And a Short Table of Integrals

Byerly, William Elwood, B. 1849

The book Elements of the Integral Calculus, With a Key to the Solution of Differential Equatons, And a Short Table of Integrals was written by author Byerly, William Elwood, B. 1849 Here you can read free online of Elements of the Integral Calculus, With a Key to the Solution of Differential Equatons, And a Short Table of Integrals book, rate and share your impressions in comments. If you don't know what to write, just answer the question: Why is Elements of the Integral Calculus, With a Key to the Solution of Differential Equatons, And a Short Table of Integrals a good or bad book?

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Suppose x is positive, and less than 1.
Then (l+^a;)""""^ approaches zero as its limit as n in- n + l creases indefinitely, for it may be thrown into the form -i- ("--^-Y^'. Since xa? ^' \l+exj zero for its limit as n increases indefinitely; as has also the factor . Hence, for values of x between and 1 , log( 1 + a;) n +1 is developable, and is equal to the series X h 2^3 4 This is true even where a? = 1 , for it is easily seen that, in that case also, ( ) approaches the limit zero as n in...- n + l \l+ffxj Digitized by LjOOQIC Chap. IX.] DEVELOPMENT IN SERIES. 129 creases. If x is between and —1, the second form of the error, Art. 128, [2], is most convenient for our purpose. Let x::s --x^ so that x' is positive and less than 1. Then our func- tion is log(l — a;') , and the series becomes ^23 n (l-(?aj')*+^ * where ^"" . is less than 1 ; hence l^n^rt /^^-^Y ^ a;' and as j is a finite value, the expression for the error de- creases indefinitely as n increases, and the function is equal to the series.

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