In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 115. [T] Recall that tan − 1 ( 1 3 ) = π 6 . Assuming an exact value of ( 1 3 ) , estimate π 6 by evaluating partial sums S N ( 1 3 ) of the power series expansion tan − 1 ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 2 k + 1 at 1 3 . What is the smallest number N such that 6 S N ( 1 3 ) approximates π accurately so within 0,001? How many term are needed for accuracy so within 0.00001?
In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum. 115. [T] Recall that tan − 1 ( 1 3 ) = π 6 . Assuming an exact value of ( 1 3 ) , estimate π 6 by evaluating partial sums S N ( 1 3 ) of the power series expansion tan − 1 ( x ) = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 2 k + 1 at 1 3 . What is the smallest number N such that 6 S N ( 1 3 ) approximates π accurately so within 0,001? How many term are needed for accuracy so within 0.00001?
In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum.
115. [T] Recall that
tan
−
1
(
1
3
)
=
π
6
. Assuming an exact value of
(
1
3
)
, estimate
π
6
by evaluating partial sums SN
(
1
3
)
of the power series expansion
tan
−
1
(
x
)
=
∑
k
=
0
∞
(
−
1
)
k
x
2
k
+
1
2
k
+
1
at
1
3
. What is the smallest number N such that
6
S
N
(
1
3
)
approximates
π
accurately so within 0,001? How many term are needed for accuracy so within 0.00001?
п/4
S tan " (x)d .
Determine a series which represents
-1
Help me
Use the fact
I a", find
that the
exponential function
the series
presentation
(using the E
notation) for
the
n=0
following
functions:
(a)
g(a).
ce
dæ
9(x) =
n=0
[The
constant C
is
already written
you).
00
h(x) =
Σ
n=0
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