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## AP®︎/College Calculus BC

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

Lesson 1: Defining convergent and divergent infinite series

# Convergent and divergent sequences

A sequence is "converging" if its terms approach a specific value as we progress through them to infinity. Get an intuitive sense of what that even means! Created by Sal Khan.

## Want to join the conversation?

• What does diverges or converges means here ?
Thank you. •   Converging means something is approaching something. Diverging means it is going away. So if a group of people are converging on a party they are coming (not necessarily from the same place) and all going to the party. Similarly, for functions if the function value is going toward a number as the x values get closer, then the function values are converging on that value.
• At , why is it (-1)^(n+1)? Is that the same as (1)^(n-1)?
Thanks • (-1)^(n-1) does equal (-1)^(n+1) for every n value. Both equations oscillate every other N, every (-1)^ to an even number equal(-1)^ to a different even number. You can change the (n-1) to any odd number [for example (n+67) or (n-7)] and get the same answer.
• I thought 1 to the infinite power was an indeterminate form, so don't you need to use L'Hopital's rule to evaluate the limit in the video? • Suppose we want to know the limit of a^b as x goes to infinity, where a and b are both functions of x. If we find that a approaches 1 and b approaches infinity, we have an indeterminate form, because we can't tell without further analysis whether the forces attracting a toward 1 (making the expression approach 1) are overpowered by the forces moving b toward infinity (making the expression approach infinity or zero, depending on whether a is slightly greater than or less than 1). So that is why we say 1^∞ is an indeterminate form. HOWEVER, if a is not some function that approaches 1, but is actually the number 1, then we no longer have an indeterminate form. The expression 1^b is always 1, no matter how large or small the exponent. We no longer have an infinitesimal increment away from 1 that can be overpowered by the increase of the exponent. So we have an indeterminate form when we have a base approaching 1 and exponent approaching infinity, but not when we have a base that EQUALS 1 and exponent approaching infinity.
• what is the difference between this statements:
a) The sequence of Bn has no limit.
b) The sequence of Bn diverges to positive infinity.
c) The sequence of Bn is simply divergent.
d) limBn=(symbol of infinity)
n--->(symbol of infinity) • I'll assume `{B(n)}` is a sequence of real numbers (but a sequence in an arbitrary metric space would be just as fine).

`a)` `{B(n)}` has no limit means that there is no number `b` such that `lim (n→∞) B(n) = b` (this may be cast in terms of an epsilon type of definition).

`b)` That `{B(n)}` diverges to `+∞` means that for every real number `M` there exists a real number `N` such that `B(n) ≥ M` whenever `n ≥ N`.

`c)` A sequence is divergent if and only if it is not convergent, hence this means the same as `a)`.

`d)` This means the same as `b)`.
• What is the difference between convergent sequence and a converging series? • is this a harmonic series? • don't all functions approach a number? like isn't infinity a number? cone somebody give me an example of a function that diverges. • No. Not all functions approach a number as their input approaches infinity. One of the main things a function has to do to approach a number is to start to stabilize. Take sine or cosine. We know they will never output anything greater than 1, or less than -1, we are even able to compute them for any real number. But, we know that they will always fluctuate. They don't head to infinity, and they don't converge. If we were to investigate sin(x)/x, it would converge at 0, because the dividing by x heads to 0, and the +/- 1 can't stop it's approach.
A similar resistance to staying mostly still can be found in equations that diverge as their inputs approach infinity. What number does 2^x go to? (It diverges)
We might claim that it goes to 34359738368, a very big number equal to 2^35, but we 2^x, being fickle to the point of cruelty, will leave this value (already large) for a notably larger one as x goes to 36. And remember, infinity is MUCH larger than 36, and this function will continue to leave potential values in the dust, never looking back, indefinitely.
Also, just in case it isn't clear, infinity doesn't follow many of the properties that numbers follow. infinity + 1 = infinity, infinity^2 = infinity / 8, basically infinity can't be dealt with in algebra, (although L'Hopitol's rule lets us manage it), and shouldn't be thought of as a "number" in the common sense of the word.
• why do we use n+1 instead of n-1 when defining the power of (-1)? is it just semantics or common protocol? I know in other cases it would be n+1 is there any reason for it not to be n-1? • What is the difference between finding the limit of a sequence and finding the limit of a function? • How did he get to the n+1 power? • We want 𝑎(𝑛) to be positive for odd values of 𝑛, and negative for even values of 𝑛.

(−1)^𝑛 is negative for odd values and positive for even values.

Multiplying by (−1) changes the sign,
so (−1)^𝑛⋅(−1) will be positive for odd values and negative for even values, which is what we want.

(−1)^𝑛⋅(−1)
= (−1)^𝑛⋅(−1)^1

So, we have the product of two powers of the same base, which means we can add the exponents, and we get
(−1)^(𝑛 + 1)