Alternating Series: Estimating the Remainder
Learn how to estimate the remainder in an alternating series by analyzing the back-and-forth behavior of the sequence. Explore the alternating pattern and understand how to calculate the sum accurately.
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Video Transcript
suppose I have an alternating series
like this one this is the sum from I
equal to 1 to infinity of minus 1 to the
I minus 1 times some other positive
decreasing sequence of limits 0 B sub I
now in my first video on alternating
series we saw that if I put down the
number line there was this sort of back
and forth behavior worthy s1 was going
to go all the way out and then the s2
was gonna go back in them yes 3 was
gonna go out and ask for back in and
that's five out of that six and s 7 and
it just bent back and forth and back and
forth like that and then we concluded
that the actual sum of this series was
going to converge in this kind of case
and it was somewhere in the middle
squash between this alternating behavior
now let me suppose I've done that and I
want to go and try to estimate this by
perhaps only these first seven terms
that is I have this series and I want to
go and write it as the sum of a partial
sum S seven and some remainder some
error term R 7 now I don't know where
exactly this sum is it could be anywhere
in this particular region but let me try
to approximate it by say the S seven so
the question is how good is this error
if I take this R 7 is it a small numbers
and a big number how can I control how
bad my estimate of only taking the first
seven terms is going to be now to think
about what this remainder could be we're
currently at the s7 and imagine we want
to go from the s7 out to an s8 well
after we went out to the s8 we would
then come back to an s9 go out to an s10
and so on but based on the way that this
collapsing works is the furthest I can
ever get away from s7 and the entire
rest of the series its gonna go to s8
because after that I'm gonna get closer
and closer and closer that's the way
this alternating series works so
whatever the remainder is I can
definitely say that that r7 is going to
be less than the distance between s8 and
s7 that is between the one term and the
