Infinite series as the limit of its sequence of partial sums

[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(x_n) be a sequence of ℝ. The sequence of partial sums (x_n) of the series ∑(x_n) is defined by s_n = ∑x_k 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.

 

[Mark44]

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.

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.

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(x_n) be a sequence of ℝ. The sequence of partial sums (x_n) of the series ∑(x_n) is defined by s_n = ∑x_k 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.

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 S_n = $\{\frac 1 2, \frac 3 4, \frac 7 8, \dots, \frac{2^{n - 1}} {2^n}, \dots\}$.
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.

 

[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.

 

[Mark44]

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.

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.

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.

Correct.