Alternating series.
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What is an alternating series? For example, this series the sum
n equals 1 to infinity of minus 1 to the n plus 1 over n,
is an example of an alternating series. Chris, what does that even mean?
Let me write down some of the terms,
because I think will be a little bit clearer.
So if I plug in n equals 1, I get negative 1 squared, which is 1 over 1.
So 1 over 1.
When I plug in n equals 2, I get negative 1 cubed over 2.
That's negative 1 over 2, so minus a half.
When I plug in n equals 3, that's negative 1 to the 4th over 3.
That's 1 over 3.
That's plus a 3rd.
When I plug in n equals 4, I get negative 1 to
the 5th, which is negative 1 over 4, so minus 1 4th.
When I plug in n equals 5,
I get negative 1 to the 6th over 5. That's 1 over 5, so plus a 5th.
When I plug in n equals 6, I get negative 1 to the
7th over 6 so, that's just negative 1 over 6, so minus a 6th.
And then it's going to keep on going like that.
But I'm flip-flopping between these two colors.
I'm alternating sines.
Right?
I'm adding, subtracting, adding, subtracting, adding, subtracting as
an alternating series. So that's what I mean by alternating.
Well here's a precise definition. The series is sum a sub
n, and goes from 1 to infinity,
is called an alternating series,
if a sub n is equal to negative
1 to the nth power times b sub n, and
all the b sub n are the same sign.
So maybe the b sub n sequence, is say all positive, and
then a sub n has this term here, times a positive sequence.
Well this term is what makes the signs, the s i g ns, flip flop back and forth.
Well here's a bit of a warning.
Not every series which has both positive
and negative terms, is an alternating series.
For example this series the sum n goes from 1
to infinity of minus 1 to the n times sign n over n.
This is not an alternating series.
I mean yes, it's got the minus 1 to the
n here, but the sine of n also introduces it's own
quite complicated pattern of positive and negative terms.
So this is a series, some of the terms are
positive, some of the terms are negative, but if not alternating.
Alright, this is not an alternating series you can cook
up other examples sum n goes from 1 to infinity.
Say, minus 1 to the n times n plus 1 over 2.
So that's all in the exponent there.
n times n plus 1 over 2,
say all over n.
This is also not an alternating series, and it's still got a minus 1
there, to raise to some power, but the power is a little bit more complicated.
n times n plus 1 over 2.
terms are some of em will be positive, some are going to be negative,
but they're not flip-flopping in sign, in s i g n back and forth.
In contrast, here's an example that is an alternating series.
the sum n goes from 1 to infinity, say of
minus 1 to the n times sine squared n over n.
Sine squared n, this turns out to always be positive.
Right?
The n is positive here, at minus 1 to the n.
This is the only thing that's affecting the s i g n of this
term, and this does indeed then flip-flop
back and forth, between negative, positive, negative,
positive, negative, positive. This is an alternating series.
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