# Infinite series as the limit of its sequence of partial sums

1. Jan 24, 2016

### Fellowroot

In my book, applied analysis by john hunter it gives me a strange way of stating an infinite sum that I'm still trying to understand because in my calculus books it was never described this way.

It says:

We can use the definition of the convergence of a sequence to define the sum of an infinite series as the limit of its sequence of partial sums. Let(xn) be a sequence of ℝ. The sequence of partial sums (xn) of the series ∑(xn) is defined by sn = ∑xk from k=1 to n.

I just need someone to explain to me how an infinite series is the limit of a sequence of partial sums.

2. Jan 24, 2016

### Staff: Mentor

I'm very surprised it wasn't in your calculus book, as what you describe is not a strange way of defining the sum of an infinite series. The usual definition of the sum of a convergent infinite series is in terms of the sequence of partial sums.
Consider this series: $\sum_{k = 1}^{\infty}a_k = \sum_{k=1}^{\infty}\frac 1 {2^k} = \frac 1 {2^1} + \frac 1 {2^2} + \frac 1 {2^3} + \dots = \frac 1 2 + \frac 1 4 + \frac 1 8 + \dots$
This is a convergent geometric series that is known to converge to 1.

The sequence of partial sums Sn = $\{\frac 1 2, \frac 3 4, \frac 7 8, \dots, \frac{2^{n - 1}} {2^n}, \dots\}$.
Here $S_1 = a_1, S_2 = a_1 + a_2, S_3 = a_1 + a_2 + a_3, \dots$

The series converges to 1 because the sequence of partial sums converges to 1. Hopefully that's clear to you.

3. Jan 24, 2016

### Fellowroot

Okay, now that I'm looking at it, its more than obvious. The red flag that got me was that it mentioned the word sequence and I didn't really get that because I always viewed this as a series not a sequence, as if they were different.

But as you have shown me, while you are calculating the series you can take those incremental sums and make a sequence out of it. The limit of the sequence should be what the infinite sum converges to. Its just another way to view the series.

4. Jan 24, 2016

### Staff: Mentor

Well, a series is different from a sequence -- I hope you understand that. A sequence is essentially a list of numbers, and a series is the sum (finite or infinite) of the terms in the series.

When you're working with a series there are two sequences to consider: (1) the sequence of terms ($a_0, a_1, a_2,$ and so on) and the sequence of partial sums ($S_0, S_1, S_2,$ and so on). In my previous post I explained where the terms in the sequence of partial sums come from.
Correct.