I Representing sums as sigma notation?

[bluejay27]

Hi is there a way or algorithm to find the sigma notation of sums in which the sums do not have an apparent general form?

 

[Mark44]

Hi is there a way or algorithm to find the sigma notation of sums in which the sums do not have an apparent general form?

To write a summation using sigma notation, you need an expression that represents the general term being added. If you don't know the general term, you're out of luck in writing a summation.

 

[bluejay27]

How about if I already have the sum and have to find the general term from it. For example, if the terms are alternating in + and -, we would have to use (-1)^n. I am struggling in finding a way to generalize the constants that are in the terms. For example if all the terms would have a 5, the general term would have a 5. If the terms are going in the progression of 1, 5, 25, and 125, the general term would have a 5^n for Maclaurin series. But these are the easier cases, I am struggling in finding the general terms for other sequences... Is there a book that will help me explain this? That directly tackles finding the general expression from the terms?

 

[mfb]

There is no general algorithm to make a useful general term for arbitrary sequences. You can always find expressions (e. g. a polynomial of suitable order), but that doesn't make them useful.

As an example, 1, 5, 25, 125 is also reproduced by $a_n = \frac {32}{3} n^3 - 56 n^2 + \frac{292}{3} n -51$. Does that make sense or help in any way? Not really. But it fits.

 

[Stephen Tashi]

I am struggling in finding the general terms for other sequences... Is there a book that will help me explain this? That directly tackles finding the general expression from the terms?

If you have a finite number of terms, there are infinitely many different formulae that will produce those finite number of terms. So, to have a well-posed mathematical problem, you have to be more specific about the type of formulae that you will consider (e.g. polynomials versus rational functions versus transcendental functions).

A squence can be considered as a special case of (x,y) data where x = the index of the term. (e.g the sequence 2, 7, 0, -3,... and be considered as (1,2) (2,7) (3,0) (4,-3),... ) Under the topic of "curve fitting" or "interpolation" you can look up methods to fit functions to curves so they pass exactly through some given (x,y) points. For example, if you want to fit a polynomial to such a sequence, you can study the lagrange interpolating polynomial ( http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html )If you have an algorithm for generating an arbitrary finite number of terms then what is mathematically known about finding a algebraic formulae that produces those terms depends on the type of algorithm. For example, in the particular case that the algorithm is a recursion like $y_{n+2} = 1 - 2y_{n+1} + y_{n}$ you can study recursion relations https://en.wikipedia.org/wiki/Recurrence_relation which are treated in books on the "Calculus Of Finite Differences" or "Difference Equations" (https://www.amazon.com/dp/0486650847/?tag=pfamazon01-20)

There can be types of algorithms where no simple algebraic formulas for reproducing their terms is known - and perhaps cases where no simple algebraic formulae exist. (https://en.wikipedia.org/wiki/Collatz_conjecture ).

 

[Stephen Tashi]

...and you can seek an expert opinion at: http://oeis.org/

 

[Stephen Tashi]

the sigma notation of sums in which the sums do not have an apparent general form?

I'm interpreting you question to mean "where the n-th term of the sequence" does not have an apparent general form.

Are you asking, instead, about how to find general formula for the the sum of n-terms of a sequence when you do know the formula for a term ?