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Surveying Sum Strategies

Date: 02/26/2011 at 12:12:35
From: Daniel
Subject: Formula for the nth term in a sum

I was wondering what general techniques there are for finding a formula
for the nth term in a sum.

For example:

    S = [?][?][i = 1, ..., n] i

You can write this out as:

    S = 1 + 2 + ... + (n - 1) + n

And as:

    S = n + (n - 1) + ... + 2 + 1

Adding the two you get:

   2S = (n + 1) + (n + 1) + ... + (n + 1) + (n + 1)

We can then see that (n + 1) appears n times and say:

   2S = n(n + 1)
    S = n(n + 1)/2

However, I'm more interested in sums that are more difficult to generalize
than this one! For example,

    S = [?][?][i = 1, ..., n] 1/i

I'm only aware of two methods for solving these:

1. This only works for polynomials (I think) and is fairly
straight-forward.

We first examine our sum; for example,

    S = [?][?][i = 1, ..., n] i^2

We then write out the first few terms (guesstimate how many are needed):

    1, 5, 14, 30, 55

We then take the first differences of consecutive terms:

    d1 = 4, 9, 16, 25

We note that the first difference values are not the same, and take the
second differences:

    d2 = 5, 7, 9

Once again our differences are not the same, so we take another difference:

    d3 = 2, 2

d3's differences are the same, which tells us that the generating
polynomial is of the third order. We then go on to solve this for specific
instances of x:

    f(x) = ax^3 + bx^2 + cx + d 

2. My second method is a lot more intuitive: it's just looking for a
pattern to formulate.

For example, given

    S = [?][?][i = 0, ..., n] 2^i

Once again, we write out the first few terms:

    1, 3, 7, 15

We notice that the values seem to be 1 less than 2 raised to some
exponent. So We go on to formulate:

    Sn = 2^(n + 1) - 1

What I find most difficult would be the lack of a "plug and play" method
approach to these problems. From what I've observed, there is no one
general approach to solving these. But that also keeps it fun!



Date: 02/26/2011 at 17:52:18
From: Doctor Vogler
Subject: Re: Formula for the nth term in a sum

Hi Daniel,

Thanks for writing to Dr. Math.

You are correct that there is no one general approach to solving a sum --
that is, to finding an explicit formula for a finite sum. But there are
methods for certain types of sums.

For summing polynomials in the index (i), see:

  Summing n^k
    http://mathforum.org/library/drmath/view/55887.html 

You probably know how to sum geometric series. You can also sum a product
of a polynomial (arithmetic) and geometric formula:

  Finding the Sum of Arithmetico-Geometric Series
    http://mathforum.org/library/drmath/view/66996.html 

This is all related to the Method of Finite Differences, which you alluded
to:

  Method of Finite Differences
    http://mathforum.org/library/drmath/view/53223.html 

For most sums, however, you cannot write the exact sum in a closed-form
formula. But for many different kinds of sums, you can get very accurate
approximations to the sum, which is often all you really want, anyway
(such as if you want to write the sum in decimal). You mentioned the
harmonic series, which is discussed at:

  Mathematical Series
    http://mathforum.org/library/drmath/view/66579.html 

Some more general formulas that use calculus to approximate the sum of certain 
types of formulas can be found here:

  Formula to Sum a Series of Square Roots
    http://mathforum.org/library/drmath/view/65309.html 

There is certainly a lot to learn in this field, although there are also many more 
questions that present-day mathematics is not able to answer.

If you have any questions about this or need more help, please write back and 
show me what you have been able to do, and I will try to offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Sequences, Series

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