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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