Understanding Series Convergence

[BarringtonT]

When I say a series $\sum$a$_{n}$ converges, what exactly is it that I am saying?
for example
$\sum^{∞}_{n=1}$$\frac{9n^{2}}{3n^{5}+5}$ is convergent. what did I just say?

 

[HallsofIvy]

Why would you say a series converges if you don't know what it means?

If you have taken a course dealing with sequences and series, then you should have seen a definition of "convergence of a sequence": the series $\sum_{n=1}^\infty a_n$ converges if and only if the sequence of partial sums $s_i= \sum_{n= 0}^i a_n$ converges.

(I hope you won't say that $\sum_{n=1}^\infty \frac{9n^2}{3n^2+ 5}$ is convergent. It obviously isn't.)

 

[BarringtonT]

Congrats on telling me exactly what the book told . so now if you don't mind tell it to me as if I was not a person studying Mathematics .

 

[mathman]

Congrats on telling me exactly what the book told . so now if you don't mind tell it to me as if I was not a person studying Mathematics .

In plain English a series is convergent if you keep adding terms of the series and it gets to a limit. For example 1 + 1/2 + 1/4 + 1/8 + ... gets closer and closer to 2 as you add more terms. On the other hand 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... keeps getting bigger as you add more terms, so it is not convergent.

 

[Bacle2]

A series converges to a value "s" if, by adding enough terms , you can get indefinitely

close to the value s. This is made rigorous : if I want to be within, say, 1/100 of the

value s, I must show that there is a term ,say "N", so that by adding N-or-more terms,

the value of the expression: (a_1+a_2+...+a_N )-s

of the sum will be within 1/100 of the value s. Take the series 1+1/2+1/4+...

Its limit is 2. After 1 term, you are within 1 unit of the limit. After adding two terms

you are within 1/2 of the limit. Now, convergence means that I can guarantee that , no

matter how close I want to get to 2, I just need to add enough terms, and my sum

will be within this --or (almost) any other--distance from 2. We do not demand that the

sum be exactly two, but that the sum be indefinitely close to it.

 

[BarringtonT]

Thank you guys very much I now understand.