prove sum, n=1 to infty, (e-(1+1/n)^n) converges or diverges

[sombra]

I've posted these two questions to the calculus forum but neither has been answered and one has been viewed 15 times and the other 9 times so I am assuming they are at a level higher than calculus even though my instructor insists they are freshman calculus.

I need to decide (prove) if the series sum(n=1 oo) (e-(1+1/n)^n) converges or diverges.

I need to test the integral (o,pi) x/sinx dx for convergence.

I have tried many things and these are the two that I have not been able to get consensus on.

If these are truly freshman calculus, how come no one can answer them but only tell me to try this or that and when I ask them to show me they don't?

Is there someone out here who would care to show me why they decide their answer? I can tell you right now that I don't think either converges but I haven't figured out how to prove either.

 

[Deleted member 4993]

Re: convergence

I've posted these two questions to the calculus forum but neither has been answered and one has been viewed 15 times and the other 9 times so I am assuming they are at a level higher than calculus even though my instructor insists they are freshman calculus.

I need to decide (prove) if the series sum(n=1 oo) (e-(1+1/n)^n) converges or diverges.

I need to test the integral (o,pi) x/sinx dx for convergence.

I have tried many things and these are the two that I have not been able to get consensus on.

If these are truly freshman calculus, how come no one can answer them but only tell me to try this or that and when I ask them to show me they don't?

Is there someone out here who would care to show me why they decide their answer? I can tell you right now that I don't think either converges but I haven't figured out how to prove either.

First of all - you need to have patience. Nobody here gets paid to spend time or answer question. So we pick and choose our problem and pick and choose our time - comprehend!!

Second - yes indeed these are freshman calculus problem.

Now you tell me the conditions for convergence - specifically an integral. Plot x/sin(x) - and look at its behavior.Then look at the conditions of convergence again and decide if it meets all the criteria?

For the second problem same routine - this time for a series. Specifically tell us - whether

\(\displaystyle \sum_{n=0}^{\infty}(1 \, + \frac{1}{n})^n\)

converges or not (This is a well-known series).

Then tell us where you are stuck - we can help you get unstuck.

 

[stapel]

Re: [MOVED] convergence of sum, n=1 to infty, of (e-(1+1/n)^n)

Duplicate threads deleted.

 

[sombra]

Re: [MOVED] convergence of sum, n=1 to infty, of (e-(1+1/n)^n)

To expand on the problems at hand: I am not allowed to use any electronic devices, calculators, graphing calculators, or integrators. I do know that e = (1+(1/n)^n). I know that in both of the problems that they diverge too, but I haven't been able to find a useable argument to show it.

I have a solution for the exponential exercise, all i need at this point is some incite for the integral (0,pi)x/sinx. I know it will increase and then decrease on that interval just by the nature of the sine function.I also can see that at pi/2 the function is about 57.3 and near pi, say at 3.14, is also 57.3 but I know it goes up and down

 

[Deleted member 4993]

Re: convergence

I've posted these two questions to the calculus forum but neither has been answered and one has been viewed 15 times and the other 9 times so I am assuming they are at a level higher than calculus even though my instructor insists they are freshman calculus.

I need to decide (prove) if the series sum(n=1 oo) (e-(1+1/n)^n) converges or diverges.



I need to test the integral (o,pi) x/sinx dx for convergence.

what happens to the value [x/sin(x)] as x ? ?

By the way [x/sin(x)] does not oscillate between 0 and ?.

Even if you are not allowed to solve the problem with calculator - you should investigate the graph within the given region to get clues regarding nature of solution.

 

[PAULK]

I've posted these two questions to the calculus forum but neither has been answered and one has been viewed 15 times and the other 9 times so I am assuming they are at a level higher than calculus even though my instructor insists they are freshman calculus.

I need to decide (prove) if the series sum(n=1 oo) (e-(1+1/n)^n) converges or diverges.

I need to test the integral (o,pi) x/sinx dx for convergence.

Here is an outline of the solution to the second:

1. Write the following infinite series, whose sum, if it exists, is this indefinite integral:

SUM integral (pi (1 - 1/2^k), to pi(1 - 1/2^(k+1))) x/sinx dx, for k = 0 to infinity.

[In other words, integrate over a set of intervals of half - decreasing length which converge to the right endpoint.

2. For each sub-integral, compare it to the 'inner-sum' rectangle, with height = the function value at the left endpoint.

3. That height approaches 2^k, while the width is pi/2^(k+1), and the product of those (the area) is pi/2.

4. So the terms of the series --> pi/2 and the sum diverges.

[Some details omitted, especially for #3.]

I must agree, however, that this is not freshman calculus, having taught that many times. Sophomore, yes.

 

[sombra]

I need to show whether the series SUM (n=1 to oo) (e-(1+1/n)^n) converges or diverges. I thought I had a solution to this problem but I've been told that just isn't so.

I think I need help understanding the difference between the nth term and the series converging. I don't seem to be able to differentiate the two. I have looked at one proof why the harmonic series doesn't converge, and the proof uses the fact that the sum is always greater than 1+n/2.

In my problem I would like to find such an animal but I am at a loss.


The following is the proof I have come up with and I've been told all I am showing is that the nth terms isn't 0 and that that isn't enough to conclude divergence. Can someone help me get this right?