If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 10

Lesson 1: Defining convergent and divergent infinite series

# Infinite series as limit of partial sums

Infinite series are defined as the limit of the infinite sequence of partial sums. Since we already know how to work with limits of sequences, this definition is really useful.

## Want to join the conversation?

• How do you come up with the formula for partial sums of non geometric series? The limit strategy is good to evaluate the sum but it all seems kind of useless if you cannot come up with the formula. • There are several ways. One is to use algebra to deduce a formula, like Gauss did with S = 1,2,3,4,...(http://mathandmultimedia.com/2010/09/15/sum-first-n-positive-integers/). Another is to use established partial sums to derive new ones, like a generalization of Gauss' method to any arithmetic series. A third way is to use induction. For example, given the series 1,3,5, ..., the partial sum looks a lot like n^2 as one does S1, S2, etc. You can then prove this inductively. First the series is 2n+1. We want to prove that n^2 = S_n, so plugging in n we see that n^2=n^2, therefore the next partial sum is the next term(2n+1) + the sum of the pervious n terms (n^2). Plugging in the next n into our partial sum formula we see that (n+1)^2 = n^+2n+1, which is what we got earlier. This shows that given a partial sum = n^2, all partial sums after that follows that pattern. Then we simply do 1+3 = 2^2 to prove that there is a partial sum = n^2. I imagine that there are more ways but those are the ways I can think of. I feel like on AP tests they will be given though.
(1 vote)
• What are the potential errors that arise if, when testing for convergence, we take the limit of the general formula of the last element in the sequence instead of the partial sum? • Let 𝑎(𝑛) denote the 𝑛th term in a series and let 𝑆(𝑛) denote the 𝑛th partial sum. we know that if lim(𝑛 → ∞) 𝑆(𝑛) converges, then the infinite sum exists and is equal to that limit. However, what if lim(𝑛 → ∞) 𝑎(𝑛) converges? What does this say about 𝑆(𝑛)? Well we know that:
𝑆(𝑛 + 1) = 𝑎(𝑛) + 𝑆(𝑛)
We can take the limit of both sides:
lim(𝑛 → ∞) 𝑆(𝑛 + 1) = lim(𝑛 → ∞) [𝑎(𝑛) + 𝑆(𝑛)]
Suppose that the infinite series exists at 𝑘 (i.e. lim(𝑛 → ∞) 𝑆(𝑛) = 𝑘). Then we have:
𝑘 = 𝑘 + lim(𝑛 → ∞) 𝑎(𝑛) ⟹ lim(𝑛 → ∞) 𝑎(𝑛) = 0
So the sum can only converge if the terms are approaching 0. This makes sense intuitively because the terms need to get smaller and smaller if we have any hope of our sum "jumping" around less and less and "settling" on a single value. If the terms converged to any nonzero number, then naturally, the sum diverges as the sum would keep going on and on.

But it should be noted that this is a necessary but not a sufficient condition for series convergence. It is possible for the terms of a series to converge to 0 but have the series diverge anyway. The classic example of this is the harmonic series:
𝚺(𝑛 = 1) ^ ∞ [1/𝑛]
Obviously here, the terms approach 0, (lim(𝑛 → ∞) 1/𝑛 = 0) but in fact, this sum diverges! So the fact that the terms of a series approach 0 is a necessary but insufficient condition for series convergence. On the other hand, the fact that the partial sums of a series converge is in fact a sufficient condition for convergence because this is exactly what we define series convergence to be. An infinite sum exists iff the sequence of its partial sums converges.

Comment if you have questions!
• In this video the teacher refers to a term and its subscript (S sub 3 for example) as the sum of the first 3 terms, but in the next video each term and its subscript are evaluated separately for a value. Which is true or is one true sometimes and not other times? • Depends on what `S` means. If you are talking about sequences and `S` represents a sequence, then `Sᵢ` will refer to the value of the i-th element in the sequence.

If you are talking about series, and `S` represents the partial sum of the series, then `Sᵢ` represents the partial sum up to the i-th item of the series.

There is nothing special about the letter `S`, you can use any symbol you want to represent either sequences or series.
• At why does Sal divide, both the numerator and denominator by n^2? • It is a method to more easily see what the end behavior will be. Hopefully you see that dividing the numerator and denominator by the same thing doesn't change the fraction.

Now, when you divide by the highest degree variable you get a bunch of terms with variables only in the denominators. When you look at a term that has a variable only in the denominator and its end behavior that term goes to 0, since one over a very large number is practically 0, and the larger ti gets the closer to 0 it gets.

This means only the terms that had degrees to match the largest degree you divided by are left. this means the function, whatever it is, becomes identical to a function with just those terms.

An example, if you look at (x^2 + x + 1)/(x^2 + 10000000x + 1000000000), if you compared the two at higher and higher x values they would look more and more like the same function. Specifically x^2 / x^2 = 1
• Shouldn't he have divided the n values by n^3 instead of n^2 because n^3 is the highest exponent? • heyy, what does s base infinity equals to? • Aren't there are nested sums with infinite limits? • If the limit of the partial sums is 0, does that mean that the sum of the series equals 0? • Yes. because the sum of the series is defined as the limit of partial sum sub n with n approaching towards infinity. As far as I know (if I'm wrong please correct me), this can only happen if either a sub n oscillates around zero with positive and negative numbers of the same magnitude somewhere in the sequence (e.g. 5 - 7 - 3 + 7 - 5 + 3 + ... = 3 - 3 + 5 - 5 + 7 - 7 + ... = 0), or the partial sum always stays at zero as a sub n is also equal to zero.
(1 vote)
• In the lesson "Worked example: sequence convergence / divergence," the fourth example was that wonderful sequence d_n = (-1)^n, the terms of which oscillate between 1 and -1. The terms of the partial series are interesting: {1, 0, 1, 0, 1 ...} ... so these oscillate as well.

My question, then, is: how can we use limit analysis to confirm that, according to convention, this oscillating set of partial sums also diverges?
(1 vote) • That's just a matter of showing that the sequence {1, 0, 1, 0...} diverges. If the limit of the sequence was L, then there would be a point in the sequence after which every term was within, say, 1/2 of L.

Every term of the sequence is either 0 or 1, so we need an L such that |1-L|<1/2 and |0-L|=|L|<1/2.
But this first inequality means that L is in the interval (1/2, 3/2) and the second one means that L is in (-1/2, 1/2). These intervals don't intersect, so no such L exists.

Since the limit doesn't exist, the sequence is divergent. 