01
Approximating
Using the series representation of
, show that: Now use the alternating series error bound to approximate
to an error within .
Solution
03
Notice that
. We have this series for
: Therefore:
Now we use the “Next Term Bound” rule. Calculate terms until we find a term less than
: So we take the following partial sum approximation:
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02
Some estimates using series
Without a calculator, estimate
(angle in radians) with an error below . (Use the error bound formula for alternating series.)
Solution
11
Write the alternating series for
: This is an alternating series, so we can apply the “Next Term Bound” rule. Calculate some terms:
(Without a calculator, we can see that
. Dividing by will only decrease this value.) So we add up the prior terms:
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03
Some estimates using series
Find an infinite series representation of
and then use your series to estimate this integral to within an error of . (Use the error bound formula for alternating series.)
Solution
12
Write the series of the integrand:
Integrate:
Now apply the “Next Term Bound” and look for the first term below
: So we simply add the first two terms:
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