What does it mean for a series to converge on a number k?

[The_Z_Factor]

In my book it is talking about sequences and series and such. Finite and infinite and all that, and I am confused with what it says in the book. The book says,

"If the terms of a finite sequence are added to obtain a finite sum, it is called a series. If a series is infinite, the sum up to any specified term is called a "partial sum". If the partial sums of any infinite series get closer and closer to a number k, so that by continuing the series you can make the sum as close to k as you please, then k is called the limit of the partial sums, or the limit of the infinite series. The terms are said to "converge" on k. If there is no convergence, the series is said to 'diverge'".

The bolded part is the part I don't exactly follow. What I don't get is how it says the partial sums get closer to a number k? What does it mean there. I may just be thinking about it the wrong way.

 

[quasar987]

It's good to look at many books at the same time, because different books explain things differently so one thing that a book is unclear about may be very clearly explained in another. Personally, I think the dfn in your book is crap.

In any case, to answer your specific question, they are saying that the limit of the sequence of partial sum is k. You see, if

\{a_i\}_{i\in\mathbb{N}}

is an infinite sequence, then for each positive integer n,

S_n=\sum_{i=1}^{n}a_i

is called the n-th partial sum of the terms in the sequence. Now, {S_n} is a sequence of numbers, and it may or may not converge. By series, we mean the value of the limit of this sequence, that we note by

\lim_{n\rightarrow +\infty}S_n=\lim_{n\rightarrow +\infty}\sum_{i=1}^{n}a_i:=\sum_{i=1}^{+\infty}a_i