infs-general

Infinite Series------General method for finding Sum out

1. Introduction

When we were a little child, we usually had some kind of "stupid" questions. Sure as is there a thing or number of adding something infinitely. For, instance, is there a number, a sum, which we add (1+1+1+1+....+ n+....) a sets of one up?? In this text, we seek to find the answer out in this text.

2. The definition of Infinite Series.

In fact we can describe those adding as Infinite Series. Symbolically,

In this series, we call the numbers as term.

3. The definition of Convergence and Divergence.

Before we point out a approach for finding the sum of a series, we need to know two following concepts:

Intuitively, without serious mathematical treatment, we definite Convergence and Divergence as:

If the result of a series is a nonfinite number, means L, then it implies the series converge

Alternatively. if the result is an infinite number, or increaseing (decreasing) without bound, then the series diverge.

However, it is impossible for one to add all terms together (as the number of term is infinite), so as to find the sum of an infinite series.

Fortunately, there is a usually feature of series for dealing this problem with the concept of convergence:

The n-th partial sum for an infinite series, says

, is

Note that, if the n-th partial sums of the series converges (diverges), then the series converges (diverges).

Also, if Sn is convergent, then we call the limit of Sn to be the sum of the series.

Instead of verify this feature with serious mathematics, in here we seek to illustrate its by a figure:

Therefore, as this figure, when there is a number n that is >k, then the feature we mentioned before is available.

So, we can answer the stupid question now,