Is there a riemann sum to integral converter?

If not then,

So I thought I could make some progress using the maple limit command in the this problem:

limit(Sum(csc(Pi*x/i)^3*sin(Pi*x)^3/i^3, i = 1 .. n), n = infinity)

which gives signum(sin(Pi*x)^3/x^3)*infinity

but what does this mean? 

How do I convert Sum(csc(Pi*x/i)^3*sin(Pi*x)^3/i^3, i = 1 .. n), n = infinity to an integral or from this result?

Its difficult for me to explain what I'm looking for so I will try with an example:

Example 1

Is there anyway any function that will say for example the sum of squares of integers for a given n say n=4, 1^2 + 2^2 + 3^2 + 4^2 = 30 that will give the equation ((n^3)/3) + ((n^2)/2) + (n/6) that gives the sum for a given n?

It would be determined from the series 1^2 + 2^2 + 3^2 + 4^2 + ... or the equation for the elements 1^2, 2^2, 3^2, 4^2 the equation that gives the sum; in this case the well known ((n^3)/3) + ((n^2)/2) + (n/6) .

So far I've got FindSequenceFunction[{1, 4, 9, 16}, n] but this only finds the individual elements not an equation for the sum of them. Of course I already know the equation for producing the individual elements.

1) I do not understand,

Ei(1, -ln(a-2)) for real numbers > +2

how to I convert it into this series format are there are 2 parameters ? (from https://en.wikipedia.org/wiki/Exponential_integral)

I'm aware of https://www.maplesoft.com/support/help/Maple/view.aspx?path=Ei&term=Ei

but as you can deduce I'm still having difficulties. 

2) Also In mary boas's book she only deals with Ei(x) no where can I figure out how 0.2193839344 is calculated from Ei(1,1.)?

I'm  trying to converting an infinite series from several terms to sigma form.

The series in question is:

I've attached the worksheet 1.mw.

The example below works but not for the above I just get:

Error, invalid arguments to coeffs

Not sure where I'm going wrong any idea how to do this?

EXAMPLE that works from https://math.stackexchange.com/questions/2786296/strategy-or-software-for-representing-infinite-series-in-sigma-form

I've also attached the worksheet for this 2.mw

=>

S:= 1-x+4/3*(x^2)-2*x^3+16/5*(x^4)-16/3*(x^5)+64/7*(x^6)-16*x^7+256/9*(x^8)-256/5*(x^9)+1024/11*(x^10);

Coeffs:= [seq(coeff(S, x, j), j=0..10)];

g:= gfun:-guessgf(Coeffs,x);

convert(g[1], FPS, x);

results in:

or 

rec:= gfun:-listtorec(Coeffs, a(n));

aa:= rsolve(rec[1],a(n));

Sum(aa*x^n, n=0..infinity);

results in the same desired result.

Hi, I'd like to use something to compare expressions, so I thought evalb or verify.

I'd like to evaluate things like the identity,

log_b(x ^y) = y ∙ log_b(x)

However using either I get:

> verify(2*log(x), log(x^2));

                             false
> evalb(2*log(x) = log(x^2));

                             false
now I thought maybe conditions might be different but as none are specified they should be the same in any domain. 

Not sure what is wrong or which comand is more suitable.