Talk:Series (mathematics)/Archive 1

In mathematics, are all infinite series indeed series of what mathematicians call terms?

This sentence, while no doubt true in some sense, makes little sense to me: "Typically, the first few terms in the infinite series nearly solve an equation." Can someone elucidate, please?----

The definition is given backward, stressing what the one of the purpose of a series is. That purpose is, given a function or constant, whose value is not known, to define another function expressed as the limit of an infinite series, where the value of the infinite series approaches the value of the original function or constant, as the number of terms increases.

Now, circling back to your question, this infinite series is picked because it is clear from evaluating it for the first few terms, that it is approaching the the value of the function or constant. Then the sentence:

"Typically, the first few terms in the infinite series nearly solve an equation." means that: When one uses an infinite series to express the value of a function or cconstant, the first few terms will approximate the value of the function rather closely, and the more terms the closer the approximation.

Note this definition is very limited. There are infinite series developed for functions whose value is already known.. For example, by a Taylor's expansion of a function or a Taylor series, a known function can be expressed by an infinite series which is a polynomial function. The purpose of finding a polynomial function or expansion of a known function is that the polynomial expansion may be a more useful expression of the original function.

In more detail:

A finite series is written a1+a2+...an, where a1,a2,..,an are called terms in the series for some positive integer n. The sum of the series is simply the sum the of n terms.

For example, it can be proved that the sum of the squares of the first n numbers, where n is a positive integer is:

          n
          Σ = 1²+ 2²+...n²=(1/6)*n*(n+1)*(2n+1)
         r=1

An infinite series is written a1+a2+a3+... , with terms a1,a2,a3,..., and one term corresponding to each positive integer. NOW, if sn is the sum of the first n terms, we look at the sequences s1,s2,s3,....

If this sequence has a limit, say, s, as n goes to infinity, the original infinite series is said to have a sum = s. If there is no such limit, the infinite series has no sum.

Clear as mud..eh?

The definition was a feeble attempt to explain an infinite series in layman terms and then give a formal mathematical definition.

A math adverse reader would be put off by the formal definition.

I know the formal definition but couldn't explain it well enough in plain words.