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Unformatted text preview: for x = 3, we have the series
This is the opposite of the alternating harmonic series, which converges. . So the convergence set is the half-closed interval (1, 3].
11. Find the Maclaurin series for sin x + cos x.
These two series are known, so all I need to do is to add them term by term. Here's what I get. 12. Find the first four nonzero terms in the power series based at a = 0 for the function
multiplying might be easier than dividing.]
For these two functions we have the series . [Hint: , which converges for all x, and , which converges for –1 < x < 1. Multiplying these gives 13. Find the Maclaurin polynomial of order three for the function y = sinh x. Then give a bound for the error in
using P3(1) to estimate sinh(1).
To investigate error, I'll need the fourth derivative anyway, so I'll make a table.
0 sinh x
1 cosh x
2 sinh x
3 cosh x
4 sinh x
So the third-order Maclaurin polynomial for y = sinh x is
P3(1) for sinh(1) is exactly equal to
This is bounded by sinh(1)/24. And the error involved in using
, for some c between zero and 1....
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- Fall '1