For wht values of p does converge

Chapter 7, Problem 4E

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QUESTION:

For what values of p does \(\int_{1}^{\infty} x^{-p} d x\) converge?

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QUESTION:

For what values of p does \(\int_{1}^{\infty} x^{-p} d x\) converge?

ANSWER:

Step 1 of 6

           

      Definition ;

                                     The    partial sum of the series  is given by

    =  +  ++..............+ . If the sequence of these partial sums {Sn} converges to L,

then the sum of the series converges to L. If {Sn} diverges, then the sum of the series diverges.

                   =     is  convergent  . Conversely  , a series is divergent if the sequence of partial sums is divergent. If    and   are convergent series , then   +  )   and  -   )    are convergent  . If  C ,  then  is convergent  series.

NOTE : The terms grow without bound ,  so the sequence does not converge.

 Integral test ; The series can be compared  to an integral to establish  convergence or divergence.

                       Let f(n) =  be a positive monotone decreasing function . If

                                         f(x) dx =   f(x) dx <  ,

then the series converges. But if the integral diverges, then the series does so as well.

 

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