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Understanding Alternate Series Estimation Theorem for Accurate Sum Approximation
Learn the Alternate Series Estimation Theorem to accurately approximate the sum of series, ensuring correct values to two decimal places. Enhance your understanding of this powerful theorem for precise calculations.
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Video Transcript
consider this problem
so let's say if we have the series of
negative one
raised to the n plus one
divided by n squared
how can we approximate the sum
of this series
correct to two decimal places
well in order to do that
we need to be familiar with the
alternate series estimation theorem also
known as the alternate series remainder
so let's talk about the basic idea of
that theorem
so let's say if we have an alternate
series
it could be negative one to the n or
negative one to the n plus 1
times a sub n
so that's the general form
now this series has to be convergent
which means that
it has to pass the divergence test
the limit as n goes to infinity of a sub
n has to be 0 and also it has to be a
decrease in sequence
the next term has to be less than or
equal to the previous term
so if you have a convergent alternating
series
then the following will be true
the difference between the infinite sum
and the partial sum
is equal to the remainder
and the remainder is less than or equal
to
a sub n plus one
so how can we use this information
to approximate this sum
correct to two decimal places
first let's make sure that we have a
convergent
series
so let's perform the divergence test
as n goes into infinity
what's going to happen to a sub n
well we need to know what a sub n is
and a sub n
is basically
one over n squared
