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- Remainder Estimate for the Integral Test: Explained with Example
Remainder Estimate for the Integral Test: Explained with Example
Learn about the remainder estimate theorem for the integral test and understand how to approximate the sum of an infinite series using the sum of the first 10 terms. Dive into the concept and calculations in this informative video.
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Video Transcript
in this video we're going to talk about
the remainder estimate theorem
for the integral tests
so let's begin with this problem
approximate the sum of the infinite
series 1 over n squared by using the sum
of the first 10 terms
so let's start with
the infinite series from one to infinity
so we're going to approximate it
by adding up the first ten terms
so basically we're looking for
s sub ten
s sub ten is going to be one over one
squared plus one over two squared
plus one over three squared
and we're going to continue this trend
up to 1 over 10 squared
so if you plug that into your calculator
you should get 1.549768
and so we could say that
the infinite series
is approximately
1.549
now granted the true answer is going to
be different than that answer but this
is simply an approximation
using the first ten terms
according to part a
so this is the answer for part a
now let's move on to part b
estimate the error with this
approximation
so how can we do that
well let's talk about the remainder
estimate for the integral tests
so let's say if we have the sequence a
sub n
and it's equal to
f of n
the function f has to be continuous
positive and decreasing
everywhere where x is equal to or
greater than n
and at the same time
the series
must be convergent
so let's check to see if these four
conditions are met
looking at the series 1 over n squared
