Integration by infinite series

if f(x) is a converging series, ff(x). dx is also convergent. Thus, if 

Series (3) is convergent when x is less than unity, for all values of n. Series (4) is convergent when TiX, and therefore when x is less than unity. The convergency of the two series thus depends on the same condition, x l. If the one is convergent, the other must be the same. 

If the reader is able to develop a function in terms of Taylor s series, this method of integration will require but few words of explanation. One illustration will suffice. By division, or by Taylor s theorem, 

The two following integrals will be required later on. k2 is less than unity. 

Develop (1 - a )-1 in series. Multiply through with log x. dx. Then integrate term by term. The quickest plan for the latter operation will be to first integrate J nlog x. dx by parts, and show that 

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