:

## Interesting Improper Integrals

A colleague showed this to me earlier this afternoon. I can explain, and accept, most of Maple's responses. I do have one case where I believe Maple could do better.

This arose during the creation of some Maple materials to support the derivation of the Integral Test for series convergence. Consider:

```restart;
I1 := Int( 1/x^p, x=1..infinity );
/infinity
|          1
|          -- dx
|           p
/1          x
```

Asking for the value of the integral, with no assumptions on p, gives:

```value( I1 );
/   (-p + 1)                  \
|  x         - 1              |
limit|- -------------, x = infinity|
\      p - 1                  /
```

Good. This result does depend on the value of p.

So, look at three different situations with assumptions on p:

```value( I1 ) assuming p>1;   # correct
1
-----
p - 1

value( I1 ) assuming p=1;   # correct
infinity

value( I1 ) assuming p<1;   # not simplified (cancel p-1 from numer and denom)
infinity p - infinity
---------------------
p - 1
```

The last result is the intriguing one. I have two questions about this:

1.
2. how does Maple get to this specific form (from I1)?
3.
4. why doesn't Maple simplify this result (under the stated assumption) to infinity?

Of course,

```simplify( % );
infinity
```

It is interesting to see how Maple responds when the improper integral is evaluated without any assumptions

```R1 := int( 1/x^p, x=1..infinity );
/   (-p + 1)                  \
|  x         - 1              |
limit|- -------------, x = infinity|
\      p - 1                  /
```

Now, applying the assumptions on p

```R1 assuming p>1;            # correct
1
-----
p - 1

R1 assuming p=1;            # incorrect (but didn't have a chance because antiderivative is incorrect)

/   (-p + 1)                  \
|  x         - 1              |
limit|- -------------, x = infinity|
\      p - 1                  /

R1 assuming p<1;            # correct
infinity
```

The result for p=1 is not surprising as the antiderivative does not apply in this case. But, Maple is able to handle the p<1 case without any trouble.

Here is the complete worksheet with these results

Looking forward to some discussion on this one.

Doug

```---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/```

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