The proof that diff(ln(1+1/n),n) < diff(R,n) for n > 0, can be also done as follows,

A:=normal(diff(ln(1+1/n),n)/diff(R,n));

                           (1/3)          (1/3) 4
                         (n      + (n + 1)     )
            A := -----------------------------------------
                    1/3        1/3         (2/3)    (2/3)
                 8 n    (n + 1)    ((n + 1)      + n     )

B:=1/n^(1/3)*((n+1)^(1/3)-n^(1/3))^4/(n+1)^(1/3)/
    ((n+1)^(2/3)+n^(2/3))/8;

                                 (1/3)    (1/3) 4
                         ((n + 1)      - n     )
            B := -----------------------------------------
                    1/3        1/3         (2/3)    (2/3)
                 8 n    (n + 1)    ((n + 1)      + n     )

normal(A-B);

                                  1

So A=1+B with B>0. That means A>1. Multiplying by the denominator, since it is negative, the sign of the inequality changes and we get the desirable inequality.

From here, ln(1+1/n) - R is decreasing for n>0, and its limit at infinity is

limit(ln(1+1/n)-R, n=infinity);

                                  0

That means that it is positive for n>0, q.e.d.

Alec

The proof that diff(ln(1+1/n),n) < diff(R,n) for n > 0, can be also done as follows,

A:=normal(diff(ln(1+1/n),n)/diff(R,n));

                           (1/3)          (1/3) 4
                         (n      + (n + 1)     )
            A := -----------------------------------------
                    1/3        1/3         (2/3)    (2/3)
                 8 n    (n + 1)    ((n + 1)      + n     )

B:=1/n^(1/3)*((n+1)^(1/3)-n^(1/3))^4/(n+1)^(1/3)/
    ((n+1)^(2/3)+n^(2/3))/8;

                                 (1/3)    (1/3) 4
                         ((n + 1)      - n     )
            B := -----------------------------------------
                    1/3        1/3         (2/3)    (2/3)
                 8 n    (n + 1)    ((n + 1)      + n     )

normal(A-B);

                                  1

So A=1+B with B>0. That means A>1. Multiplying by the denominator, since it is negative, the sign of the inequality changes and we get the desirable inequality.

From here, ln(1+1/n) - R is decreasing for n>0, and its limit at infinity is

limit(ln(1+1/n)-R, n=infinity);

                                  0

That means that it is positive for n>0, q.e.d.

Alec

As I said (below), for this question you need just to find the first term of asymptotic of (1+1/n)^a - e,

asympt((1+1/n)^(n+a)-exp(1),n,2);

                     exp(1) (- 1/2 + a)      1
                     ------------------ + O(----)
                             n                2
                                             n

From here, the asymptotic is O(1/n) for any a except a=1/2, for which the asymptotic is O(1/n^2), which is much better (i.e. the error is smaller.)

Alec

As I said (below), for this question you need just to find the first term of asymptotic of (1+1/n)^a - e,

asympt((1+1/n)^(n+a)-exp(1),n,2);

                     exp(1) (- 1/2 + a)      1
                     ------------------ + O(----)
                             n                2
                                             n

From here, the asymptotic is O(1/n) for any a except a=1/2, for which the asymptotic is O(1/n^2), which is much better (i.e. the error is smaller.)

Alec

For example, evaluate q for a=1, b=4. That gives 3/4 while a*b^(1/2) would give 2.

For such things, the normal Maple input (1D), is much easier to use, and easier to correct the input errors.

Alec

That might be a good question for a poll - whether people prefer collaborative editing, or more or less the same as it is currently - with only moderators being able to edit other people posts. I think, giving "high-ranked users" moderator status might serve better this site needs, but other people might think different.

The current poll doesn't seem especially important - whatever settings are in the browser, they could be easily changed if necessary.

Alec

I'd like to be able to edit my posts even if somebody replied to them.

Stack Overflow has the shared editing - anybody (well, not anybody - so called "trusted users") can edit anybody else's posts. That's something questionable, I think.

Also, it looks quite ugly and has advertisements, same as Yahoo and other sites mentioned in this context. I hope this won't be implemented here.

Personally, I prefer Silverlight forums and MSDN forums look and feel.

Alec

I thought somebody said something about modern look. Flash is outdated even more than Drupal.

Modern look = Silverlight

Alec

Shareholders decided to sell the company, and Cybernet made the best offer.

Alec

There are security problems with opening unknown .mw files. In particular, related to autoexecution.

Alec

Maple Worksheet - Error

Failed to load the worksheet /maplenet/primes/worksheet/11436_Algebra expresion 2.mw .

 

In the expression you posted, a lot of multiplication signs are missing. It should be a*b instead of ab and the same with parentheses, say a*(1+b) instead of a(1+b). Also, you have to be sure that the division signs are correct. For instance, /b/a is the same as /(a*b) in Maple.

Alec

In Student:-Calculus1, the variables are assumed to be real.

By the way, Maple gives the following answer (outside of Student:-Calculus1 package),

value(J);

                                    2 1/2
                            1/2 x (x )

simplify(%);

                                 2
                            1/2 x  csgn(x)

Alec

By the way, replacing abs(x) with sqrt(x^2) produces a wrong answer in ShowSolution,

J:=Int(sqrt(x^2),x);
                                /
                               |    2 1/2
                         J :=  |  (x )    dx
                               |
                              /

Student:-Calculus1:-ShowSolution(J);

                              /
                             |    2 1/2
                             |  (x )    dx
                             |
                            /

[rewrite, (x^2)^(1/2) = x]

                                   /
                                  |
                               =  |  x dx
                                  |
                                 /

[power]

                                     2
                                    x
                                 = ----
                                    2

Alec

The latest version of rtf and the previous one (implemented in MS Office 2007) displays math expression quite nicely (in a way similar to LaTeX.) and the native conversion between that and MathML is implemented in Windows 7.

If Maple could switch to that instead of currently used scheme in Standard, the copying and pasting would be much easier to implement properly, at least in Windows.

Alec