8.7 Taylor and Maclaurin series
Example 1
Find the Maclaurin series of the function f(x)= and its radius of convergence.
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R=
Example 3
Find the Taylor series for f(x)=e at a=2.
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Example 4
Find the Maclaurin series for sin(x) and prove that it represents sin(x) for all x.
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Example 5
Find the Maclaurin series for cos(x).
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Example 7
Find the Maclaurin series for f(x)=(1+x) ,where k is any real number.
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Example 8
Find the Maclaurin series for the function f(x)=
and its radius of convergence.
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2.The graph of f is shown.
(a)
Explain why the series
1.6-0.8(x-1)+0.4(x-1) -0.1(x-1)
is not the Taylor series of f center at 1.
f(1)>0,
(x-1)
+0.8
-0.8
(b)
Explain why the series
2.8-0.5(x-2)+1.5(x-2) -0.1(x-2)
is not the Taylor series of f center at 2.
f(2)<0,
(x-2)
-1.5
+1.5
3.
for n=0,1,2,...,and find the Maclaurin series for f and its radius of convergence.
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4.Find the Taylor series for f centered at 4 if
What is the radius of convergence of the Taylor series?
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Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series
expansion. Do not
show that Rn(x)-->0.]
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Use the binomial series to expand the function as a power series. State the radius of convergence.
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Use a Maclaurin series derived in this section to abtain the Maclaurin series for the given function.
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Evaluate the indefinite intergral as an infinite series.
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46.
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Find the sum of the series
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