The limit of R_n as n-->inf is...

It seems that I have some issues with the sums...

I have the function

I need to evaluate it by using the definition of Riemann sum for - right end-points,

then,

...

I'm not sure I did right so far, and how should I proceed further...

thanks for hints,

dokrbb

Re: The limit of R_n as n-->inf is...

Quote:

Originally Posted by

**dokrbb** It seems that I have some issues with the sums...

I have the function

I need to evaluate it

by using the definition of Riemann sum for

- right end-points,

then,

The sum is over i not n.

At this point you have already done the sum. You should not have summation sign.

Quote:

I'm not sure I did right so far, and how should I proceed further...

thanks for hints,

dokrbb

Re: The limit of R_n as n-->inf is...

Re: The limit of R_n as n-->inf is...

Quote:

Originally Posted by

**dokrbb**

someone???

Re: The limit of R_n as n-->inf is...

Quote:

Originally Posted by

**dokrbb** someone???

Could you post what the exact question is, as you found it? To compute from first principles would involve deriving a closed form for this partial sum:

which seems to be very difficult, I've got nothing...

Re: The limit of R_n as n-->inf is...

Quote:

Originally Posted by

**FelixFelicis28** Could you post what the exact question is, as you found it? To compute

from first principles would involve deriving a closed form for this partial sum:

which seems to be very difficult, I've got nothing...

I'll try, the question is like this:

The following sum

is a Right Riemann sum for the definite integral

where b = ...

and f(x) = ...

I checked my answer for and and they are correct; The next question is: the limit of these Riemann sums as n-> infinity is....

does that change my approach?

Re: The limit of R_n as n-->inf is...

Quote:

does that change my approach?

Yes, this changes everything.

Quote:

Originally Posted by

**dokrbb** I'll try, the question is like this:

The following sum

is a Right Riemann sum for the definite integral

where b = ...

and f(x) = ...

We use the definition of the Riemann sum to be

Clearly:

as you've done.

And , thus which you've also done correctly.

Quote:

The next question is: the limit of these Riemann sums as n-> infinity is....

When they ask for this, they're not asking you to compute this integral by evaluating this limit:

But rather, what they're asking you to do is evaluate this limit:

by using this relation:

where .

In other words, they're not asking you to compute the integral by using the Riemann sum, they're asking you to compute the Riemann sum by using the integral.

Re: The limit of R_n as n-->inf is...

Quote:

Originally Posted by

**FelixFelicis28** Yes, this changes everything.

We use the definition of the Riemann sum to be

Clearly:

as you've done.

And

, thus

which you've also done correctly.

When they ask for this, they're not asking you to compute this integral

by evaluating this limit:

But rather, what they're asking you to do is evaluate this limit:

by using this relation:

where

.

In other words, they're not asking you to compute the integral by using the Riemann sum, they're asking you to compute the Riemann sum by using the integral.

I'm not sure I understand really how it should be done, do I really need in this case to evaluate it as

and so on, without using the actual values of and ?

Re: The limit of R_n as n-->inf is...

Quote:

Originally Posted by

**dokrbb** I'm not sure I understand really how it should be done, do I really need in this case to evaluate it as

and so on, without using the actual values of

and

?

Hmmm? I'm not quite sure what you're doing...

Your nth Riemann sum is:

.

And they're asking you to evaluate .

We proceed with the definition that:

.

Thus:

.

All that's left to do is compute the integral on the RHS and you're done.